Number 24

Twenty Four (24)

1. It is the smallest number to have 8 different factors. These factors are – 1, 2, 3, 4, 6, 8, 12 and 24.

2. A day has 24 hours.

3. It is a Harshad number. A number is known as Harshad number if it is divisible by the sum of its digits. 24 is exactly divisible by 2 + 4 = 6. This number was discovered by the Indian mathematician D R Kaprekar.

4. The sum of squares of first 24 natural numbers is a square number. It is the only integer except 1 with this property.
1^2 + 2^2 + --- + 24^2 = 4900 = 70^2

5. It is one of the kissing numbers in fourth dimensional space. A kissing number is defined as the number of non overlapping unit spheres that can be arranged such that each touch another given unit sphere. Kissing numbers are also known as Newton number.

6. Unlike English alphabets, Greek alphabets have 24 letters.

7. It is the product of first four consecutive numbers.
24 = 1 x 2 x 3 x 4

8. It is the first number which can be expressed in the form of 2^3q, where q is an odd prime.
24 = 2^3 x 3

9. Puzzle: - Take any four consecutive numbers. Multiply them and the answer will be divisible by 24. Example: - Let us take the four consecutive numbers 34, 35, 36 and 37.

Its product is 34 x 35 x 36 x 37 = 1585080 which when divided by 24 gives 66045 as quotient.

10. It is the product of second triangular number and second cube numbers.
24 = 3 x 8

11. It can be partioned in 1575 ways.

12. It is the sum of 4 consecutive odd numbers.

24 = 3 + 5 + 7 + 9

13. It is the fourth abundant number.

14. It is the third nonagonal number. A nonagonal number is a polynomial number that represents a nonagon. The nth nonagonal number is given by – n (7n – 5 ) / 2.

15. It is the sum of twin prime 11 and 13.
24 = 11 + 13

16. It is the largest integer that is divisible by all natural numbers less than its square root. The square root of 24 is 4.89 approximately therefore it is divisible by 1, 2, 3 and 4.

17. For any prime number greater than 3, p^2 – 1 is divisible by 24. If we take p = 7 then 7^2 – 1 = 48 is divisible by 24.

18. The atomic number of Chromium is 24.

19. The purest form of gold is 24 carat.

20. The 24th dimension is the highest dimension for which the exact "kissing number", the number of spheres that can be placed around a central sphere so that they all are touching it, is known. For the 24th dimension, the "kissing number is 196,560. Beyond the fourth dimension, only the eighth and twenty-fourth are known exactly.  (From the blog of Pat Ballew) 


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Number 23

                            Twenty Three (23)

1. It is a prime number and more interestingly it’s both digits are also prime so is its sum. 2 and 3 are prime and 2+ 3 = 5 is also a prime.

It is a Cousin prime with 19 (differ by 4) and a sexy prime with 17 (differ by 6).

2. Biorhythm theory proposed by Wilhelm Fleiss states that men’s physical and emotional characteristics follow a 23 day cycle.

3. It is the smallest prime number with two consecutive digits.

4. In a group of 23 people, the probability of at least 2 persons having same birthday is more than 50%.

5. It is the smallest odd prime which is not a twin prime.

6. 1 repeated 23 times gives a prime number.
11, 111, 111, 111, 111, 111, 111, 111 is a prime number.

7. 23 balls can be arranged in two different ways.

8. The factorial of 23 has exactly 23 digits.

23! = 25,852,016,738,884,976,640,000

9. It is the sum of three consecutive primes 5, 7 and 11. It is the first prime number with this characteristic. 23 = 5 + 7 + 11

10. The sum of first 23 prime numbers is divisible by 23. The sum of first twenty three primes is 874

11. David Hilbert had presented 23 problems to the International Congress of Mathematicians in 1900. These problems were unsolved by that time and it was first published by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society in 1902.

12. It is the atomic number of Vanadium.

13. It is the fifth Sophie Germain prime. A prime number p is a Sophie Germain prime if 2p + 1 is also prime.

This is named after French mathematician Marie Spphie Germain. The first few Sophie Germain Primes are

–2, 3, 5, 11, 23, 29, 41, 53, 83, --

23 is a prime and 2 x 23 + 1 = 47 is also a prime.

14. It can be partioned into 1575 ways.

15. The 23rd number of English Alphabet is W and in a normal computer board W lies between 2 and 3.

16. In English alphabet 23rd letter W has two downwards and 3 upward move.

17. Trivia :- 

•Aryabhata wrote his first book ARYABHATIYA at the age of 23.

•Johann Bernoulli wrote the manuscript of De Calculo differentatium at the age of 23.

•Ramanujan got his first article published in the journal of Indian mathematical society at the age of 23 and the problem was a nested radical roots

18. It is a Happy Number. A number is called Happy Number if sum of squares of its digit after repeated operation comes out to 1.

Be happy, 23 is a Happy Number

19, 23! (Factorial of 23) has only 23 digits and it is the only number following this properties. The factorial expansion has all digits from 0 to 9.

                  23!=25,852,016,738,884,976,640,000

20. Enjoy this beautiful expression

2 ! + 3 ! = 2^3

1! + 2 x 2! +3 x 3! = 23

You can watch the movie Number 23 on the given link

https://www.youtube.com/watch?v=uTIXSKBxf-0

Happy reading 

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Dr. Rajesh Kumar Thakur

Twenty two

Twenty two (22)

1. It is a Smith number for base 10. A number is a Smith number if the sum of its digits equals to the sum of digits of its prime factors.

22 = 2 x 11 and 2 + 2 = 2 + 1 + 1

Basically, the number had its origin from a Telephone number of Harold Smith. His telephone number was 4937775 and its factors are 3 x 5 x 5 x 65837

  

4937775 = 3¹ 5² 65837¹

while

4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 · 1 + 5 · 2 + (6 + 5 + 8 + 3 + 7) · 1 = 42

2. If you cut a circle with just six line segments, the maximum number of pieces that can be so created is 22.

3. Twenty two is the fourth pentagonal number. The series of the Pentagonal number is 1, 5, 12, 22, 35…

4. There are 22 ways of linking five hexagons together having each shapes are different.

5. It can be portioned into 1002 ways.

6. 22 is a Palindrome number so is its square. The square of 22 is 484 which when read from left to right or vice versa is same.

7. It is the smallest number which can be expressed as the sum of 2 primes in 3 ways.

22 = 3 + 19 = 5 + 17 = 11 + 11

8. It is the number of writable regular polygon in a Euclidean circle.
These are 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360.

9. Factorial of 22 has 22 digits.


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Importance of Twenty One

Twenty one (21)
 1. There are twenty one dots on a die. In a die numbers are marked from 1 to 6 and they add up to 21. 1 + 2 + 3 + 4 + 5 + 6 = 21 

2. It is the product of two numbers 3 and 7 associated with good luck. 21 = 3 x 7

3. It is a triangular number.

 4. It is a Fibonacci number. The number in Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ---

 5. It is the third star number. Star numbers can be represented by a square with a triangle on each side.

Here a 3 x 3 square is surrounded by 4 triangles.

 6. If a square number ends with xy xy xy xy xy then xy will be either of these- 21, 29, 61, 69 or 84. 

508 853 989  x 508 853 989 = 258 932 38 21 21 21 21 21

 7. It is the smallest Fibonacci number whose digits are Fibonacci number and whose sum of digit is also a Fibonacci number.

 8. The atomic number of Scandium is 21.

 9. The unique projective plane of order 4 has 21 points and 21 lines, with 5 points on each line and 5 lines through each point.

10. It is a Harshad Number.

11.  21 repeated twenty-one times, following 1, forms a smoothly undulating palindromic prime
121212121212121212121212121212121212121 is prime

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Importance of 20

Twenty (20)

1.      It can be expressed as the sum of first four even numbers.                                                              

  20 = 2 + 4 + 6 + 8

2.      A polygon having 20 sides is called Icosahedron. An Icosahedron is a solid with equilateral triangles on each side of its 20 faces.                              

                                             

          

3.      It is the sum of first six Fibonacci numbers.                       

                                                                20 = 1 + 1 + 2 + 3 + 5 + 8

4.      Magical triangle with numbers from 1 to 9 can be constructed with its magical constant as 20.

                                                                                                                                     

5.      The Mayans had a base 20 counting system, subdivided into base 20.

6.   It can be partioned in 627 ways

7.      Base 20 numeral system is called Vigesimal system.

8.      It can be written as the product of a squared prime and a prime.                                                      

              20 = 2² x 5

9.      It is a tetrahedral number.                                        

                                                                                       

10.  The atomic number of Calcium is 20.

11.  After 19, it is the only number with more than one digit that can be written on base 2 to base 20 using only the digits from 0 to 9.

12.  It is the second semi perfect number because it is the sum of some of its own factors.              20 = 1 + 4 + 5 + 10

13. The 20th palindromic prime (929) has a digit sum  ( 9 + 2 + 9 = 20 ) of 20.

14. It is the sum of first four triangular number

                                   20 = 1 + 3 + 6 + 10 
15. It is a self number
   ( Read more about self number on 
http://numberbyrajeshkumarthakur.blogspot.in/2018/01/self-number.html )

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Importance of 19

Nineteen (19)

1.      It is a Prime number.

2.      It is a maximum number of 4^th powers needed to sum to any numbers.

3.      It is the smallest prime that can grow 8 times by the right.   

                                                                     

4.      It can be portioned in 490 ways.

5.      It is a hex number. It can be arranged into pattern of nested hexagon. 

  7, 13, 19… are the example of hex number.                                                                                                     

                                Hex number = 6 x Triangular Number + 1     

6.      It is the first number with more than one digit that can be written from base 2 to base 19 using only the digits from 0 to 9.      

7.      19 can be written as the sum of two consecutive numbers. Moreover, it is the difference of square of these consecutive numbers.                                                                                                        

                               19 =  9  +  10                                                                                                               
                              19 = 10² – 9²

8.      The atomic number of Potassium is 19.   

9.      It is the third number whose decimal reciprocal is of maximum length. The period of its repeated decimal is 18.                                                                                                           

                           
                           1/19 = 0.05263 15789 47368 421
10.  It is the only Palindrome number in Roman Numeral
                                   

11. It is the smallest centered triangular number.

12. 19 is also the smallest base ten number that is NOT a palindrome in any base 

                                           2≤b≤10  

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Importance of 18

Eighteen (18)

1.      It is a Lucas number. Like Fibonacci number, in Lucas number the next number of this series is obtained by summing up the previous two numbers. The series goes like- 1, 3, 4, 7, 11, 18…

2.It is the sum of four consecutive numbers.        

                                                                                 18 = 3 + 4 + 5 + 6

3.      The number 18 is considered a number of the devil due to the fact that 6 + 6 + 6 = 18 and 666 is the number of beast.

4.      It is the only number which is twice the sum of its digits.                                                           

      18 = 2 x ( 1 + 8 )

5.      It is a Pentagonal Pyramid number because it is the sum of three Pentagonal numbers.

6.      It is the second abundant number. A number is called Abundant if the sum of its divisor excluding itself is more than itself. The divisors of 18 are 1, 2, 3, 6, and 9.                                     1 + 2 + 3 + 6 + 9 > 18

7.      It can be portioned in 385 ways.

8.      The cube and fourth powers of 18 use all the digits from 0 to 9.                                                18³ = 5832                                  18⁴ = 104976

9.      It is equal to sum of the digits of its cube, its 6^th power and 7^th power.                   

 18³ = 5832 and 5 + 8 + 3 + 2 = 18                                                                                     

18⁶ = 34012224 and 3 + 4 + 0 + 1 + 2 + 2 + 2 + 4 = 18                          

                                 

18⁷ = 612220032 and 6 + 1 + 2 + 2 + 2+ 0 + 0 + 3 + 2 = 18

10.  It is the amount of time taken by Moon to tilt its orbit in order to rotate a full circle.

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