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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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