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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com