Number Zone by Dr Rajesh Kumar Thakur

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)…

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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 --