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
rkthakur1974@gmail.com

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
rkthakur1974@gmail.com

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
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