Mathematically 75

Seventy Five (75)

1. It is also called three quarter. 3/ 4 = 75%

2. It is one less than the sum of squares of its digit.
75 = 7^2 + 5^2 – 1

3. It can be expressed as the sum of 10 consecutive primes beginning from 3.
75 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

4. It is a nonagonal number.

5. It is a pentagonal pyramid number. A pentagonal pyrammdal number is a figurarive number that represents the number of objects in a pyramid with pentagonal base.
75 = 1 + 6 + 18 + 40

6. The atomic number of Rbenium is 75.

7. It is a Keith number. A Keith numbers run like a Fibonacci sequence. 75 has its digit 7 and 5. If we start to form a sequence with the digit 75 and go on finding the next number by adding the previous two as done in Fibonacci sequence we get 75.
7, 5 , 12, 17, 29, 46, 75.. It is named after Mike Keith who discovered this number in 1987

8. It is a Self-number. A self-number is an integer which in a given base cannot be generated by any other integer added to the sum of that other integer’s digit.

Example:- 21 is not a self-number as it can be generated by 15 and sum of tis digit 1 and 5, whereas 20 is a self-number as it cannot be generated by any such numbers.

9. A diamond anniversary is celebrated after every 75 years.

10. The Proper divisor of 75 are - 1, 3, 5, 15, 25 whose sum is a perfect square
1 + 3 + 5 + 15 + 25 = 49 = 7^2
Moreover , its product
1 x 3 x 5 x 15 x 25 = 5625 = 75^2
is also a square.

11. 2^75 + 75 is a prime


Dr Rajesh Kumar Thakur

Matheamatically 74

Seventy Four (74)

1. The maximal number of regions into which a plane can be divided by 9 circles are 74.

2. It is the product of two distinct prime.
74 = 2 x 37

3. It is the sum of four consecutive numbers.
74 = 17 + 18 + 19 + 20

4. The atomic number of Tungsten is 74.

Regards

Dr Rajesh Kumar Thakur

Mathematically 73

Seventy Three (73)

1. It is the 21st prime number. It along with the next prime 73 makes a twin prime.

2. It is the smallest number besides 1 which is 1 less than the double of its reverse.
73 = 2 x 37 – 1

3. It is the smallest number with the property that its neighbours can be written as the sum of two squares.
72 = 6^2 + 6^2
73 = 3^2 + 8^2
74 = 5^2 + 8^2

4. It is the smallest prime congruent to 1 modulo 24.
73 ≡ 1 (mod 24)

5. On base 8 it is the only number with repeated unit.
73 = ( 111 ) 8

6. The square root of 73 is 8. 544.. which when summed up gives 21 that is the product of its digit.

(73)^1/2 = 8. 544 and 8 + 5 + 4 + 4 = 7 x 3

7. It is a centered 12 gonal and 24 gonal number.

8. It is the least number of sixth powers needed to represent every possible integer.

9. The atomic number of Tantalum is 73.

10. There is a very interesting puzzle whose final result comes out to be 73.
The steps of the puzzles are--
-a) Take any four digit number
b) Write this number again making a eight digit number
c) Divide this 8 digit number by 137
d) divide the quotient by the 4 digit number taken initially.
The final result will be 73.
Example: - Let the four digit number taken is 1238.
Make this number an eight digit number by writing this number twice, i.e. 12381238.
Divide this big number by 137 and you get the quotient 90374 which when divided by 1238 gives you 73.

Dr Rajesh Kumar Thakur
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Mathematically 72

Seventy Two (72)
1. It is equal to half of the gross.
½ x 144 = 72

2. The sum of first eight even number is 72.
2 + 4 + 6 + 8 +10 + 12 + 14 + 16 = 72

3. It is the only number which is eight times the sum of its own digits.
72 = 8 x (7 + 2)

4. It is the smallest number that can be written as the sum of two prime number in two distinct ways and the most interesting part of this partition is that the primes used ends with 1.
72 = 11 + 61
= 31 + 41

5. The normal human pulse rate is 72 beats/ minutes at rest.

6. An isosceles triangle having a pair of 72 degree angle is called Golden triangle.

7. It is one- fifth of a central angle (360 degree) of circle.

8. The internal angle of a regular pentagon is 72 degree.

9. It is divisible by every number from 1 to 10 apart from 5, 7 and 10.

10. It is the smallest 5th power equal to the sum of 5 other 5th powers.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5

11. In financial circle rule of 72 is a simple method to quickly estimate compound interest. If you want to know how long will it take to double your money, divide 72 by the interest rate and the answer will be the number of years.

12. It is the sum of four consecutive primes as well as the sum of six consecutive prime number.
72 = 13 + 17 + 19 + 23 (sum of 4 consecutive numbers)
= 5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive numbers)

13. It is a Harshad number on base 10.

14. The atomic number of Hafnium is 72.

15. It is an Abundant number because the sum of factors of 72 excluding itself is more than 72. The factors are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

16. It is a 25- gonal number.

17. It is the maximum number of sphere that can touch another sphere in a lattice packing in 6 dimensions.

18. The great Rhombicuboctahedron in an Archimedean solid is 72 edges.

19. The sum of the eight row of Lozanic triangle is 72.
72 = 1 + 4 + 12 + 19 + 19 + 12 + 4 + 1
The Lozanic triangle is a triangular array of binomial coefficients similar to Pascal’s triangle. It is named after Sima Lozanic, the Serbian chemist.


Dr Rajesh Kumar Thakur
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Mathematically 71

Seventy one (71)

1. It is a 20th Prime number which makes a twin prime with the next prime 73.

2. It is a Chen Prime. A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. It is named after Chen Jingrun. There are infinitely Chen primes and the first ten are- 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

3. It is also a Pillai prime named after the Indian mathematician Subbayya Sivasankaranarayana Pillai. A Pillai prime is a prime number p for which there is no integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. The first 10 Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109 and 137. Mathematically

4. It is a 5th centered heptagonal number. A centered heptagonal number is a centered figurative number that represents a heptagon with a dot in the centre and other dot surrounding in heptagonal layers. It is genereated by using the formula

5. The cube of 71 contains all odd numbers from 3 to 11.
71^3 = 357911

6. The cube of 71 is the only two digit cube that ends with 11


7. The atomic number of Lanthanide is 71.

8. It is one of the number in the last pair of Brocard number. Pairs of numbers (n ,m) that solves the Brocard’s problem which states to find the integral value of n for which this was posed by Henri Brocard in 1876 and later solved by Srinivasa Ramanujan independently in 1913 without knowing that this problem had been solved earlier. There are only three pairs of such number- (4, 5), (5, 11) and (7, 11).

9. Sum of all previous prime numbers is divisible by 71.
2 + 3 + 5 + 7 + 11+ --- + 67 is divisible by 71.

10. The square of 71 can be expressed the sum of digits of its factorial.
71^2 = 7! + 1!


DR RAJESH KUMAR THAKUR
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Mathematically 70

                                                                  Seventy (70)

1. The sum of divisors of 70 including itself is a square number.
70 has the factors- 1, 2, 5, 7, 10,14, 35 and 70 which when added gives 144 which is a square number. 1 + 2 + 5 + 7 + 10 + 14 + 35 + 70 = 144 = 12^2

2. It is the smallest weird number. A number is called weird if it is an abundant number without being the sum of any set of its divisors. As discussed above that 1, 2, 5, 7, 10, 14 , 35 and 70 are the factors of 70 whose sum excluding 70 is 74 but no set of these factors sum to 70.
1 + 2 + 5 + 7 + 10 + 14 + 35 = 74
35 + 14 + 10 + 7 + 5 = 71
35 + 14 + 10 + 7 + 5 = 63

There are very few Weird number and you will be astonished to know that there are only 7 weird number below 10000 and they are 70, 836, 4030, 5830, 7192, 7912 and 9272.

3. The sum of squares of first 24 natural numbers is the square of 70.
1^2 + 2^2 + 3^2 + … + 24^2 = 70^2

4. It is a Pell number. Pell numbers are similar to Fibonacci numbers and are generated by the formula
An = 2 An – 1 + An – 2
The first 10 Pell numbers are 1, 2, 5, 12, 29, 70, 169, 408, 985 and 2378.

5. In base 10 it is a Harshad number as it is divisible by sum of its digit.

6. It is the seventh pentagonal number.

7. The atomic number of Yttterbium is 70.

8. 2⁷⁰ = 1180591620717411303424. The sum of the digits is also 70, 

DR RAJESH KUMAR THAKUR
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Mathematically 69

                                                              Sixty Nine (69)

1. It is the fifth number that stays same when written upside down.

2. It is the only number whose square and cube together use all the digits from 0 to 9.
69^2 = 4761
69^3 = 328509

3. 6 and 9 when rotated with 180 degree gives the same number. Moreover, the binary value of 6 and 9 are complement to each other.
(6)10 = 110 = (0110)2
(9)10 = (1001)2.

In binary number system 0 and 1 are complementary to each other here the complement of 0110 is 1001 which is nothing but the number 6 and 9 written in decimal system.

4. 10^69 + 69 and 10^69 - 69 is also prime number.

DR RAJESH KUMAR THAKUR
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Matheamatically 68

Sixty Eight (68)

1. It is the smallest composite number that becomes prime by turning it upside down. 68 turning upside down gives 89 which in a prime number.

2. It is the sum of 8 consecutive numbers beginning from 5.
68 = 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

3. It is a Truncated Tetrahedral number.

4. There are exactly 68 ten digit binary numbers in which each digit is the same as one of it's adjacent digits.

5. The atomic number of Erbium is 68.

6. 68 is the largest known number to be the sum of two primes in exactly two different ways:
                         68  =7+61
                              =31+37.

7. In a normal distribution around 68% of value are within one standard deviation away from the mean.

DR RAJESH KUMAR THAKUR
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Mathematically 67

                                                                        Sixty Seven (67)

1. It is the smallest number which is both palindromic in base 5 and 6.
(67)10 = (232)5 = (151)6

2. The sum of 5 consecutive prime numbers starting from 7 is 67.
7 + 11 + 13 + 17 + 19 = 67

3. The cousin prime of 67 is 71. Two primes that differ by 4 is called Cousin Primes.
(3, 7), (7, 11), (13, 17)... etc. are cousin primes

4. The atomic number of Homium is 67.

5. It is the maximal number of regions into which 11 lines divides a plane.

6. It is a centered 11- gonal and 22- gonal number.

7. A prime number with a 7 in it form an interesting pattern.
67 x 67 = 4489
667 x 667 = 444889
6667 x 6667 = 44448889

8. More interesting fact
67=(110)+(111)+(112)
      = 1 + 11 + 55

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DR RAJESH KUMAR THAKUR
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Mathematically 66

Sixty Six ( 66 )

• It is the sum of first 11 natural numbers.
1 + 2 +….+ 11 = 66

• It is a triangular number.

• There are 66 blocks in the given skeleton tower. This tower is six blocks high and has been built with 66 blocks.

(Courtesy :- Richard Philips :- Number)

• It is a Palindrome number.

• It is a Hexagonal number.

• It can be written as the product of three primes.
66 = 2 x 3 x 11

• The atomic number of Dysprosium is 66.

• 66 has seven divisors including itself. If you sum up all these divisors then you get a square number.

1 + 2 + 3 + 6 + 11 + 22 + 33 + 66 = 144 = 12^2

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 65

Sixty Five (65)

1. It is the product of two odd primes.
65 = 5 x 13
2. It is the lowest integer that becomes a square if its digit reversed is either added or subtracted from it.
65 – 56 = 9 = 3 x 3
65 + 56 = 121 = 11 x 11

3. The difference of square number of 65 and its reverse 56 is also a square number and the trio forms a Pythagorean triplets.
65^2 – 56^2 = 33^2

4. In the ASCII code used in computer 65 represents A.

5. It is the fifth star number.

6. It is a Rhombic Dodecahedral number.

7. It is a octagonal number

8. The atomic number of Terbium is 65.

9. The numbers of Euler’s idoneal numbers are 65. In mathematics, Euler's idoneal numbers, are the positive integers D such that any integer expressible in only one way as x^2 ± Dy^2 (where x^2 is relatively prime to Dy^2) is a prime, prime power, or twice one of these. The 65 idoneal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848

10. It is the length of the hypotenuse of 4 different Pythagorean triangles.
65^2 = 16^2 + 63^2
= 33^2 + 56^2
= 39^2 + 52^2
= 25^2 + 60^2

11. It can be expressed uniquely in power of 5 consecutive numbers with 5 consecutive powers in reversed order.
65=15+24+33+42+51

12. It is a magic constant of 5 x 5 magic square.

13. The sum of 10 natural numbers starting from 2 is 65.
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =65
14. The sum of numbers in each ring of the concentric circle in the given magic circle is 65.

9 + 23 + 12 + 1 + 29 = 65
2 + 16 + 10 + 24 + 13 = 65
6 + 17 + 3 + 14 + 25 = 65
4 + 15 + 21 + 7 + 18 = 65
11 + 5 + 19 + 8 + 22 = 65

DR RAJESH KUMAR THAKUR
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Mathematically 64

Sixty Four (64)

1. It is the smallest number with exactly 7 divisors. The divisors of 64 are 1, 2, 4, 8, 16, 32 and 64.

2. It is the second number after 1 which is a square number, a cube number or the sixth power of a number.
64 = 2^6 = 8^2 = 4^3

3. The difference of first pair of amicable number is 64. The first pair of amicable number is 220 and 284.
284 – 220 = 64

4. It is also the difference of 10th pair of amicable number. The 10th pair of amicable number is 66992 and 66928.
66992 - 66928 = 64

5. The number of codons of DNA in a human body is 64.

6. The atomic number of Gadolinium is 64.

7. There are 64 squares in a chess board.

8. It is the sum of first 8 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

9. It is a dodecagonal number.

10. It is the number of plus (+) signs needed to write the partitions of 8.

11. There are 64 hexagrams in I Ching. It is the one of the world’s oracle and most loved revered book of Chinese wisdom. This book is consulted for its advice and insight into human nature and telling the future.

12. It is the smallest number with 6 primes.
64 = 2 x 2 x 2 x 2 x 2 x 2

13. It is the sum of 4 consecutive centered hexagonal numbers.
64 = 1 + 7 + 19 + 37


DR RAJESH KUMAR THAKUR
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Mathematically 63

Sixty Three (63)

1. It is a Harshad number divisible by sum of its digits.
63 ÷ (6 +3) = 7

2. On base 4 , it is a repeated digit number.
63 =(333)4

3. It is a 22 gonal number.

4. It is a Woodall number, named after H J Woodall. Numbers in the form of n x 2n – 1 is called Woodall number.
63 = 4 x 24 – 1

5. The atomic number of Europium is 63

6. It is a Kaprekar number for 2 digits. If we take any two digit number and reverse it , take the difference of two and repeat the process we get 63 at the end.
53 – 35 = 18 81 – 18 = 63

7. It is a Dealnnoy number. A Dealonnoy number D describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east. It is denoted by-

Dr Rajesh Kumar Thakur
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Mathematically 62

Sixty Two (62)

1.July and August each having 31 days in a month adds up to 62, the highest number of days in consecutive months.

2. A Truncated icosahedron has 62 faces with a mixture of decagons, hexagons and squares.

3. It is the smallest number that can be expressed as the sum of 3 distinct squares in 2 distinct ways.

62 = 1^2 + 5^2 + 6^2
= 2^2 + 3^2 + 7^2

4. There is no number which is 62 times the sum of its digits.

5. It is the product of two prime numbers.
62 = 2 x 31

6. The atomic number of Samarium is 62.

7. In base 5, it is a repeated digit number.

8. It can be expressed as the sum of four consecutive numbers.
62 = 14 + 15 + 16 + 17

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 61

Sixty One (61)

1. It is the smallest prime whose reverse 16 is a square.

2. It is the smaller prime such that its square can be expressed as the sum of 1, 2,3 and 4 distinct squares.

61^2 = 60^2 + 11^2
= 52^2 + 24^2 + 21^2
= 56^2 + 22^2 + 10^2 + 1^2

3. The 61st Fibonacci number (2, 504, 730, 781, 961) is the least Fibonacci number that contains all the digits from 0 to 9.

4. It is a hex number. If you have 61 coins you can arrange it into a hexagonal pattern with one coin in the middle.

5. The reciprocal of 61 = 1/61, has the decimal period of 60 which includes 6 occurrence of each of the digits 0 to 9.
1/61 = 0.0169442622950819672111---- It is the smallest reciprocal whose period has this property.

6. It is the 5th hexagonal number.

7. It is the 9th Mersenne prime exponent.
261 − 1 = 2,305,843,009,213,693,951

8. It can be expressed as the sum of squares of two consecutive numbers.
61 = 5^2 + 6^2
9. It is the 8th Keith number which moves like the Fibonacci sequence where the next term is obtained by adding the previous two terms.
6, 1, 7, 8, 15, 23, 38, 61---
10. The atomic number of Promethium is 61

Mathematically 60

Sixty (60)

1. The Babylonian used the sexagesimal number system (base 60) to write numbers.

2. It can be divided by all numbers from 1 to 6 and also by 10, 12, 15, 20, 30 and 60. The Sumerians used this property to divide the sky into 6 parts each divided into units of 600.

3. It is the smallest number to have 12 factors—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

4. The internal angle of equilateral triangle is 60degree.

5. The product of first Pythagorean triplet is 60.
3 x 4 x 5 = 60

6. It is the sum of twin prime 29 and 31.
60 = 29 + 31

7. It is a Harshad number because it is divisible by the sum of its digit.

8. The icosahdron has 60 equivalent edges.

9. It is a Heptagonal Pyramidal number.

10. The atomic number of Neodymium is 60.

11. In one hour there are 60 minutes and in 1 minute there is 60seconds.

12. Archimedian solids – truncated icosahedrons, rhombicosidodecahedron and snub dodecahedron have 60 vertices
13. It is a 21- gonal number.

14. It is an abundant number.

15. It is the sum of four consecutive primes.
60 = 11 + 13 + 17 + 19

16. It is the smallest number that can be expressed as the sum of two odd primes in 6 ways.
60 = 7 + 53
= 13 + 47
=19 + 41
= 17 + 43
= 29 + 31
= 23 + 37

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 59

Fifty Nine (59)

1. It is the smallest number which on division by 2, 3, 4, 5 and 6 leaves remainder 1, 2, 3, 4 and 5 respectively.

2. For every 59 rotations of Earth , Mercury rotates once.

3. It is one of the solution of the number which says –a^4 + b^4 = c^4 + d^4 where a< b < c < d. This was solved by Euler in 1772. 59^4 = 133^4 + 134^4 – 158^4 4. Only 59% of the moon’s surface can be observed from the earth. 5. It is the 17th smallest prime number. 6. Icosahedron has 59 Stellations. Stellation is a process of constructing new polygons in 2 dimensions and new polyhedral in 3 dimensions.

7. It lies in the middle of 3 x 3 prime magic square with magical constant as

17 89 71
113 59 5
47 29 101

8. It is one of the factors that divide the smallest composite Euclid number. Euclid numbers are integers of the form En = pn# + 1 where pn# is the product of first n primes. 13# + 1 = 30031 = 59 x 509 is the first composite Euclid number.


Dr Rajesh Kumar Thakur

Mathematically 58

Fifty Eight (58)

1. It is the sum of first seven prime numbers.
2 + 3 + 5 + 7 + 11 + 13 + 17 = 58

2. It is the sum of four consecutive numbers.
13 + 14 + 15 + 16 = 58

3. It is a Smith number. Smith number is a composite number, the sum of whose digits is the sum of digits of its distinct prime factors.
58 has two factors 2 and 29 58 = 2 + (2+9)

4. It is an 11- gonal number.

5. It is a centered 19- gonal number.

6. The atomic number of Cerium is 58.

7. It is the minimum wind speed mile/h needed to issue a severe thunderstorm warning.


Read More about Numbers in my book THE POWER OF NUMBERS available on flipkart, amazon and other online book selling companies

Dr Rajesh Kumar Thakur
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Mathematically 57

Fifty seven (57)

1. It is a 20 gonal number.

2. It can be expressed as the sum of -- x^y + y^x where x and y are integers.
2^5 + 5^2 = 57

3. It is the sum of digits from 7 up to 12.
7 + 8 + 9 + 10 + 11 + 12 = 57

4. It is the product of two primes.
57 = 3 x 19

5. On base 7 it is a repeated digit.
57 = (111)7


Dr Rajesh Kumar Thakur

Mathematically 56

Fifty Six (56)

1. It is a tetrahedral number. It is the sum of first six triangular numbers.
1 + 3 + 6 + 10 + 15 + 21 = 56

2. It is an abundant number.

3. It is a tetranacci number. Like Fibonacci sequence, in the tetranacci numbers the next number is the sum of previous four terms. The first few tetranacci numbers are---
0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773—

4. It is the maximum number of regions into which 10 lines can divide a plane.

5. Here is an interesting puzzle involving 56.
111 + 1 / 1 + 1 = 56 222 + 2 / 2 + 2 = 56
333 + 3 / 3 + 3 = 56 444 + 4 / 4 + 4 = 56

6. The atomic number of Barium is 56.

7. The number of reduced 5 x 5 Latin square is 56. Here is an example---
1 2 3 4 5 2 3 5 1 4 3 5 4 2 1 4 1 2 5 3 5 4 1 3 2


Dr Rajesh Kumar Thakur

Mathematically 55

Fifty Five (55)

1. It is the sum of first 10 natural numbers.
1 + 2 + --+10 = 55

2. It is a triangular number. It is the first triangular number composed of repeated digits.
The other are 66 and 666.

3. It is the largest triangular number in Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

4. The sum of squares of first five natural numbers is 55. The sum of squares of n natural number is given by --- 1^2 + 2^2 + 3^2 +---- + n^2 = n (n+1) (2n+1) /6
1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55

5. It is a Palindromic number.

6. Every integer greater than 55 is the sum of distinct primes of the form 4n + 3.

7. The square of 55 shows an interesting pattern.
55^2 = 3025 and 30 + 25 = 55

8. It is the product of 3rd and 5th prime numbers.
55 = 5 x 11

9. Sum of 8 factors of 36 is 55.
1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55

10. It is a square pyramidal number.

11. It is a centered nonagonal number.

12. It is a centered 18 - gonal number.

13. It is the 4th Kaprekar number.

14. It is a cubic recurring digital invariant. Add the cubes of its digits together and repeat the operation twice and you will get 55 back.
55 : 5^3 + 5^3 = 250
250: 2^3 + 5^3 + 0^3 = 133
133: 1^3 + 3^3 + 3^3 = 55

15. It is a heptagonal number. A polygonal number of the form of n(5n – 3 ) / 2 is called heptagonal number.

16. If you multiply any odd numbers from 91 to 109 by 55, you will get a palindrome number each time.
91 x 55 = 5005 93 x 55 = 5115

17. We all know
                           3^2 + 4 ^2 = 5^2
but                      33^2 + 44^2 = 55^2

Dr Rajesh Kumar Thakur

Mathematically 54

Fifty Four (54)
1. It is an abundant number.

2. It is the smallest number that can be written as the sum of 3 squares in 3 different ways.
54 = 7^2 + 1^2 + 2^2 = 6^2 + 3^2 + 3^2 = 5^2 + 5^2 + 2^2

3. It is the sum of first 9 natural numbers starting from 2.
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 54

4. There are 54 coloured squares on a standard Rubik’s cube. Each of 6 faces has 9 squares. This was invented by Hungarian Sculptor Erno Rubik in 1974.

5. It is a Leyland number. A Leyland number is written in the form of x^y + y^x where x and y are integers greater than 1. The first few Leyland numbers are --- 8, 17, 32, 54, 57---

6. The atomic number of Xenon is 54.

7. It is a Harshad number because it is divisible by the sum of its digits.

8. It is a 19- gonal number.

Dr Rajesh Kumar Thakur

Mathematically 53

Fifty Three (53)


1. It is a Prime number.

2. If there are 53 people in a room it is very likely that at least two of them share the same birthday.

3. It is the only two digit number which on reverse changes into a hexadecimal base.
53 = 3516
4. The sum of all consecutive primes from 7 to 53 is the product of 7 and 53.
7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37+ 41+ 43 + 47 + 53 = 371 = 7 x 53

5. It can be expressed as the sum of squares in two different ways.
53 = 22 + 72 = 12 + 42 + 62

6. The sum of 53 prime numbers is divisible by 53.
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 173 + 179 + 181 + 191 + 197 + 199 + 211 + 223 + 227 + 229 + 547 = 5830 = 53 x 110

7. The sum of first 53 natural numbers is also divisible by 53.

1 + 2 + 3---- + 53 = 1431 = 53 x 27

8. It is the sum of five consecutive primes starting from 5.
5 + 7 + 11 + 13 + 17 = 53

9. It is the smallest prime which is not the difference between powers of 2 and 3.

10. Atomic number of Iodine is 53.

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 52

Fifty Two (52)

1. The number of playing cards is 52. It contains 4 suits (Heart, Diamond, Clubs, and Spades) each of 13 in numbers. The 52 cards have 26 Black and 26 Red Cards. 4 kings, 4 queens, and 4 ace are known as ace cards.

2. It is the sum of two consecutive primes.
52 = 23 + 29

3. A piano has 52 white keys.

4. It is the sum of 6 Fibonacci number starting from the 3rd.
52 = 2 + 3 + 5 + 8 + 13 + 21

5. A year has 52 weeks.

6. It is a 6th Bell number. Bell numbers are named after the mathematician Eric Temple Bell. It is the number of partitions of a set with n members or the number of equivalence relation on it. The first 6 Bell numbers are –
1, 1, 2, 5, 15, 52---

The Bell numbers can be generated by the following table, which works kind of like Pascal's Triangle.

7. It is a decagonal number.

8. It is a 3rd Untouchable number. An untouchable number is a positive integer that can’t be expressed as the sum of all the proper divisors of any positive integer including the untouchable number itself. The first few untouchable numbers are 2, 5, 52, 88---

9. The atomic number of Tellurium is 52.


Dr Rajesh Kumar Thakur

Mathematically 51

Fifty One (51)

1. It is a Pentagonal number as well as Centered Pentagonal number.

2. It is the smallest number which can be written with the digits 1 to 5 as the sum of prime numbers. 51 = 2 + 3 + 5 + 41

3. It is the sum of 6 numbers in an AP with common difference 3.
51 = 1 + 4 + 7 + 10 + 13 + 16

4. It is equal to sum of 6 numbers from 6 to 11.
51 = 6 + 7 + 8 + 9 + 10 + 11

5. It is a 18- gonal number.

6. It is the product of two primes.
51 = 3 x 17

7. It is a 6th Motzkin number. For a given number n, Motzkin number is the number of different ways of drawing chords on a circle between n points. It is named after Theodore Motzkin. The first few Motzin numbers are- 1, 2, 4, 9, 21, 51-----

8. The atomic number of Antimony is 71,

Dr Rajesh Kumar Thakur

Mathematically 50

Fifty (50)

1. It is the smallest number that can be expressed as the sum of two squares in 2 different ways.

50 = 1^2 + 7^2 = 5^2 + 5^2
2. It is equal to the sum of squares of 3 consecutive numbers.
50 = 3^2 + 4^2 + 5^2

3. It can be written as the sum of consecutive prime numbers in two different ways.
50 = 3 + 11 + 13 + 23
= 2 + 5 + 7 + 17 + 19

4. It can be written as the product of complex number and its conjugate.
50 = (7+ i) x ( 7 – i)

5. The atomic number of Tin is 50.

6. It can be written as the sum of digits involving 0 to 9.
50 = 19 + 43/ 2 + 76/5
= 28 + 36/4 + 91/7
= 39 + 21/7 + 64/8

7. It is the sum of squares of first Pythagorean triplet 3 – 4 – 5
50 = 3^2 + 4^2 + 5^2

8. It is a hexagonal pyramid number. It is a pyramidal number formed by adding the first few hexagonal numbers. The first few numbers are – 1, 7, 22, 50..

Dr Rajesh Kumar Thakur

Mathematically 49

Forty Nine(49)
1. It is a square number.
49 = 7 x 7

2. It is the sum of first 7 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49

3. It is the smallest square whose digits can be separated to make two other squares.
7^2 = 49 = 4 / 9 13^2 =169 = 16 / 9
19^2 = 361 = 36 / 1 35^2 = 1225 = 1 / 225

4. Its cube ends with 49.
49^3 = 117, 649

5. If you put 48 in the middle of number 49 and its successive results, a square number will be formed each time.

4 48 9 = 67^2
44 48 89 = 667^2
444 48 889 = 6667^2

6. The reciprocal of 49 is a repeating decimal with a period of 42.
1⁄49 = 0.0204081632 6530612244 8979591836 7346938775 51

7. The atomic number of Indium is 49.

8. It is a centered octagonal number.

9. The square of 49 = 2401 whose sum 2 + 4 + 0 + 1 =7 is a square root of 49.


Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 48

Forty Eight (48)

1. It is the smallest number with exactly 10 divisors. Its divisors are- 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

2. It is the smallest even number that can be expressed as a sum of two primes in 5 different ways.
48 = 5 + 43 = 11 + 37 = 7 + 41 = 17+ 31 = 19 + 29

3. It is equal to number of hours in 2 days.

4. The square of 48 can be expressed as the sum of squares of 9 terms of an AP of common difference 3.
48^2 = 2^2 + 5^2 + 8^2 + 11^2+ 14^2 + 17^2 + 20^2 + 23^2 + 26^2

5. It is a Harshad number. Harshad number is divided by the sum of its digits.
48 ÷ 12 (4 + 8) = 4

6. The product of all proper divisors of 48 is equal to 48^4.
48^4 = 1 x 2 x 3 x 4 x 6 x 8 x 12 x 16 x 24 = 5308416

7. The atomic number of Cadmium is 48.

8. It is an abundant number.

9. It is a 17- gonal number.

10. The Rhombicuboctahedron is an Archimedan solids with 48 edges.

Dr Rajesh Kumar Thakur

Mathematically 47

Forty Seven (47)

1. It is a prime number.

2. It is a Lucas number. The Lucas series goes like – 1, 3, 4, 7, 11, 18, 29, 47—

3. It is the first of three consecutive primes (47, 53, 59) that are spaced an equal distance apart. The first such set is (3, 5, 7) and next set is (151, 157, 163)

4. It is the largest number of cubes that cannot tile a cube.

5. The combination of 2 makes an interesting pattern.
47 + 2 = 49 and 49 x 2 = 94
497 + 2 = 499 and 497 x 2 = 994
4997 + 2 = 4999 and 4997 x 2 =9994
49997 + 2 = 49999 and 49997 x 2 = 99994
--------so on
the above pattern shows the same digits in reverse order in two operations.

6. It is the difference of squares of two consecutive numbers.
47 = 24^2 – 23^2

7. There are 47 occurrences of 47 in the first thousand prime numbers.

8. The atomic number of Silver is 47.

9. It is a carol number. A carol number is an integer of the form of 4n – 2n+1 – 1. The first few number are – 1, 7, 47, 223--- This was discovered by Cletus Emmanuel who named it after his friend Carol G Kimon.


Dr Rajesh Kumar Thakur

Mathematically 46

Forty six (46)

1. It is the product of two primes.
46 = 2 x 23

2. It is a centered Triangular number.


3. It is a nonagonal number.


4. It is a centered 15 gonal number.

5. It can be partitioned in 89134 ways.

6. It is the maximal number of regions into which 9 lines divide a plane.


7. The atomic number of Palladium is 46.


8. The number of chromosomes in every cell of human body is 46. 23 of them are inherited from mother and 23 from father.

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 45

Forty Five (45)

1. It is a Kaprekar number. A number n is called Kaprekar number if n^2 = xy and x + y = n.
45^2 = 2025 and 20 + 25 = 45

2. It is a triangular number.

3. It is the sum of first nine natural numbers. The formula to sum the n natural number is given by – n (n+1)/2. 1 + 2 + 3+ 4 + 5 + 6+ 7 + 8 + 9 = 45

4. It is the sixth positive integer with a prime factorization in the form of p2q, with p and q are primes.
45 = 3^2 x 5

5. The atomic number of Rhodium is 45.

6. It is the 5th hexagonal number that can be obtained using the formula – n(2n – 1). The hexagonal numbers are – 1, 6, 15, 28, 45---

7. It is a name of card game played between two players as two teams. The winner is the first to gain 45points.

8. It can be partitioned in 75175 ways.

9. It is a 16- gonal number.

10. Here is a mystic rose pattern that contains 45 lines. It is drawn by placing 10 marks around a circle and joining them.

Dr Rajesh Kumar Thakur

Mathematically 44

Forty four (44)

1. It is the sum of 8 consecutive numbers.
2 + 3 + 4 +5 + 6 + 7 + 8 = 44

2. It is a palindromic number of two digits.

3. It is a Tribonacci number. The Tribonacci numbers are somewhat like Fibonacci numbers with a difference that here the next number is the sum of preceding terms and it starts with 0.
The first few Tribonacci numbers are—0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81—

4. It is an octahedral number. It is a figurative number that represents the number of spheres in an octahedron formed from closed packed spheres. The first few octahedral numbers are--- 1, 6, 19, 44, 85, ---. The nth octahedral number is given by --

5. The atomic number of Ruthenium is 44.

6. It can be partitioned into 63261 ways.

7. Along with number 45, it is the first pair of consecutive numbers with 6 divisors. The sequence of such pairs are- (75, 76), (98, 99), (116,117), (147,148)…


Read More about Numbers in my book - THE POWER OF NUMBERS Published by PRABHAT PRAKASHAN

you can buy this book from FLIPKART, AMAZON, GOOGLEBOOK, SNAPDEAL etc

Dr Rajesh Kumar Thakur

Mathematically 43

Forty Three (43)
1. It is a Prime number.

2. It forms a twin prime along 41. There are infinite twin primes.

The list of first ten twin primes are---
(3,5), (5,7), (11,13), (17.19), (29.31), (41,43) , (59,61), (71,73), (101,103), (107,109)

3. It is the smallest non palindromic prime which on subtracting its reverse gives a perfect square.
43 – 34 = 9 = 32

4. It is the smallest number which can be expressed as the sum of primes in different ways.
43 = 2 +41
= 3+17+ 23
=11 +13 +19
=2 + 5+ 7 +29
= 2 +11 +13 +17

5. It can be partitioned into 53174 ways.

6. It is the smallest prime that can grows 5 times by the right.
43 is prime
439 is prime
4391 is prime
43913 is prime
439133 is prime

7. It is the smallest prime which is not a Chen prime. A prime p is called Chen prime if p+2 is either a prime or a product of two primes. It is named after Chen Jingrun. The first few Chen Primes are- 2,3, 5,7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47—

8. It is the fourth term of Sylvester’s sequence.
43 = 2 x 3 x 7 + 1

9. The atomic number of Technetium is 43.

10. It is a centered Heptagonal number. A centered heptagonal number is a centered figurative number that represents a heptagon with a dot in the center and other dots surrounding the center dot in successive heptagonal layers. The formula for n centered heptagonal number is given by-

11. It is a centered 14 - gonal number.

12. It can be partioned in 53174 ways.


Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 42

Forty two (42)

1. It is the 8th abundant number

2. It lies between twin prime numbers 41 and 43.

3. It is the smallest perfect square that is the mean of two cubed primes.
422 = (113 + 133 ) / 2

4. It is the sum of first 6 even numbers.
42 = 2 + 4 + 6 + 8 + 10 + 12

5. n^2 – n is divisible by 42.

6. The 3 x 3 x 3 magic cube can be constructed with the help of numbers from 1to 27 such that the sum of every row, column and diagonal is 42.

7. It can be partitioned into 44583 ways.

8. It is a repdigit in base 4.
42 = (222)4

9. The atomic number of molybdenum is 42.

10. It is the product of first three terms of Sylvester’s sequence. Sylvester’s sequence is named after the mathematician James Joseph Sylvester. Sylvester’s sequence is an integer sequence in which each member of the sequence is the product of the previous members plus one. The first five members of Sylvester’s sequence is --- 2, 3, 7, 43 and 1807.

42 = 2 x 3 x 7

11. It is the reciprocal of sixth Bernoulli numbers (1/42) discovered by great Swiss mathematician Jakob Bernoulli. The first few Bernoulli numbers are--
B0 = 1, B1 = ±1⁄2, B2 = 1⁄6, B3 = 0, B4 = −1⁄30, B5 = 0, B6 = 1⁄42, B7 = 0, B8 = −1⁄30.

12. It is a Catalan number. The series for Catalan numbers are--- 1, 1, 2, 5, 14, 42, 132, 429---.Catalan numbers are probably the most frequently occurring combinatorial numbers, after the binomial coefficients. It is named after the Belgian mathematician Eugene Charles Catalan. The nth Catalan number is defined as --

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 41

Forty one (41)

1. It is the 5th centered square number.

2. It is a prime number

3. It is the sum of two consecutive squares.
41 = 4^2 + 5^2

4. It is the lowest number whose cube can be expressed as the sum of cubes of 3 numbers in two different ways.
41^3 = 2^3 + 17^3 + 40^3
= 6^3 + 32^3 + 33^3

5. It is the smallest non- palindromic prime which on subtracting its reverse gives a perfect cube.
41 – 14 = 3^3

6. It is the sum of first six prime numbers.
41 = 2 + 3 + 5 + 7 + 11 + 13

7. It is a centered square number. It is a centered figurative number that gives the number of dots in a square with a dot in the centre and all other dots surrounding the center dot in successive square layer. The first few centered square numbers are 1, 5, 13, 25, 41 etc.

8. It can be partitioned in 37338 ways.

9. x2 + x + 42 gives a prime number for all x from 0 to 39. It was discovered by Euler.

10. It is the largest known number n for which the decimal expansion of 11n does not contain any zero.

11. It is the smallest prime whose sum of the digits and difference of digits are twin primes. 4 + 1 = 5 and 4 – 1 = 3 are twin primes.

12. It is the smallest prime of the smallest set of three consecutive primes {41, 43, 47} whose sum of digits (5, 7, 11) is also a set of three distinct primes.

13. It is the sum of first 3 factorials plus the first three self powers.
1! + 2! +3! +1^1 + 2^2 + 3^3 = 41

14. It is the first number of a prime number sequence that is 41 numbers long. In other words, if you start from 41 and add 2 , 4, 6, 8, 10, --- etc you will have a string of 40 straight prime numbers.
41 + 2 = 43 43 + 4 = 47 47+6 = 53 etc.


15. It is the smallest number that is not in the form of 2^x – 3^y

16. It is the smallest prime such that the sum of factorials of its distinct digits is a square number.

4! + 1! =5^2

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 40

Forty (40)
1. It is the only number whose letters are in the alphabetical order.
FORTY

2. It can be partitioned into 31185 ways.

3. It is a pentagonal pyramid number being the sum of first four pentagonal numbers. The first few pentagonal pyramid numbers are – 1, 6, 18, 40, and 75. The formula for the nth pentagonal pyramid number is given by--

4. It is the 7th abundant number.

5. The Celsius and Fahrenheit temperature come together at minus 40.
- 40degree C = - 40degreeF

6. One Furlong is equal to 40 rods.

7. It is an octagonal number. An octagonal number is a number that creates a octagon. The first few octagonal numbers are 1, 8, 21 and 40.

8. It is the fourth star number. This pattern is made up of a square and four equilateral triangles.

Dr Rajesh Kr Thakur
rkthakur1974@gmail.com

Mathematically 39

Thirty Nine (39)

1. It is the sum of five consecutive prime numbers.
39 = 3 + 5 + 7 + 11 + 13

2. It is the product of two odd primes.

39 = 3 x 13

3. It can be partioned in 26015 ways.

4. It is the sum of 10th odd and 10th even number.
39 = 19 + 20

5. It is the sum of first three powers of 3.
39 = 3^1+ 3^2 + 3^3

6. It is the sum of numbers 4 through 9.
39 = 4 + 5 + 6 + 7 + 8 + 9

7. It is the sum of 6th, 7th and 8th odd numbers.
11 + 13 + 15 = 39

8.It (39) is the smallest number with multiplicative persistence 3. [Multiplicative persistence is the number of times the digits must be multiplied until they produce a one digit number; 3(9)= 27; 2(7) = 14; 1(4)=4. ] - 

9. The F26A graph is a symmetric graph with 39 edges. In graph theory, the F26A graph is a symmetric bipartite cubic graph with 26 vertices and 39 edges.

10. It is equal to the number of plus (+) signs needed to write the partitions of 7.

11. It is the smallest natural numbers which has 3 partitions into three parts (25, 8, 6), (24, 10, 5) and (20, 15, 4)-which all give the same product when multiplied.

12. It is a 14- gonal number or Tetradecagonal number. The formula to obtain a 14 gonal number is ½ (12n^2 – 10n).

13. The largest Armstrong number in decimal numbers has 39 digits. (115,132,219,018,763,992,565,095,597,973,522,401 is the largest)

A number is called Armstrong number if it can be expressed as the sum of the its own digits , each raised to the power  of the number of digits of the number. It is also called Narcissistic number.

There are just four numbers, after unity, which are the sums of the cubes of their digits: , , , and 407 = 4^ 3 + 0^3 + 7^3

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematically 38

Thirty Eight (38)

1. The only possible magic hexagon has the magical constant as 38. The following magic hexagon has the digits from 1 to 19.

2. It is the largest known even number which can be represented as the sum of two distinct primes in one and only one way.

38 = 31 + 7

3. It is the sum of squares of first three primes.

38 = 2^2 + 3^2 + 5^2

4. 38th digit of pi (excluding decimal) is a Prime number
 31415926535897932384626433832795028841 

5. It is the largest even number which cannot be written as the sum of two odd composite numbers.

6.

5. The atomic number of Strontium is 38.

6. It is the product of two prime factors.
38 = 2 x 19

Dr Rajesh Kumar Thakur

Mathematically 37

Thirty Seven


1. It is the fourth Hexagonal number, because it can be arranged in a pattern of nested hexagon.

2. It is a prime number.

3. Normal body temperature of a healthy man is 37 degree C

4. If you multiply 37 with the multiple of 3 you will get an astonishing result.
37 x 3 = 111
37 x 6 = 222
37 x 9 = 333
37 x 12 = 444 etc.

5. Take any 6 digit number in which the first three digits are consecutively increasing and last three digits are consecutively decreasing, the number will be divisible by 37.
123987 is divisible by 37

6. It is the only two digit number which, when multiplied by 2 and subtracted by 1 gives the number in reverse order.
2 x 37 – 1 = 73 is reverse of 37

7. The atomic number of Rubidium is 37

8. 37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits.

9. The reciprocal of 37 has three periods.
1/37 = 0.027 027 027 …
It is the only prime that has three digit periods after decimal expansion.

10. It is a centered 12- gonal number.

11. It is a maximal number of regions into which 8 lines divide a plane.

12. It can be partioned in 21687 ways.

Dr Rajesh Kumar Thakur