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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Importance of Number 17

Seventeen (17)

1.      Seventeen can be poartioned in 297 ways.

2.      Studies have shown that if you ask the people to pick a number from 0 to 20, 17 will come most of time. That’s why it is also known as psychologically random number.

3.      It is the 7^th Prime number and only prime of the form – p^q + q^p, where p and q are primes.                   
                   17 = 2³ + 3²

4.      It is the only prime which is the sum of four consecutive prime.                                            
                  17 = 2 + 3 + 5 + 7

5.      It is the lowest number which can be written as A³ + B² in two distinct ways.                     
       17 = 2³+ 3²                                                                                                                      
                          = 1³ + 4²

6.      It is the third Fermat Prime. 

.                                   17 = 2⁴ + 1

7.      C F Gauss (1777- 1855) has discovered a way of accurately drawing a regular heptagon (17 side polygon) by using compass and ruler.

8.      There are exactly 17, two dimensional space groups known as wallpaper groups because they represent the 17 possible symmetry types that can be used for wallpaper.

9.      It is also known as Feller number after the name of famous mathematician William Feller. It is said that while discussing unsolved mathematical problem with his students he would say to try this problem for an arbitrary value of n = 17          

10.  The atomic number of Chlorine is 17.

11. It is the sum of two distinct fourth powers.     

                                                                                   17 = 1⁴ + 2⁴

12.  It is the only prime with the property that it is equal to the sum of digits of its cube.                 17³ = 4913 and its digit sum = 4 + 9 + 1 + 3

13.  It lays exactly half way between 16 which is a square and the number 18 which is divisible by a square number 9. These two numbers 16 and 18 have the distinction that the areas and perimeter are equal. The area of square and perimeter is 16 while the area and perimeter of the rectangle is 18.

14. There are 17 Prime partition of Number 17.

15. If you write out the numbers from 1 to 5000 in English (e.g., THREE THOUSAND EIGHT HUNDRED SEVENTY-THREE), it turns out that 17 is the only one that has a unique number of characters (nine). (Spaces and hyphens count as characters) .

15. The sum of square of all primes up to 17 is a beast number.

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Importance of Number 16

                                               Sixteen 

1.      It can be portioned into 231 ways.

2  It is also a fourth power and a square number.                                                                         

    4² = 2⁴ = 16

3. It is the sum of first four odd numbers.                                                                                      1 + 3 + 5 + 7 = 16

4. It is the only integer with property x^y = y^x where x ≠y                                                               4² = 2⁴

5.      It is the smallest number with exactly 5 divisors 1, 2, 4, 8 and 16.

6.      A number with base 16 is called Hexadecimal. The 16 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

7.      It is the first square number that can be expressed as the sum of two triangular numbers in 2 different ways.         

                                                  

                                            16 = 1 + 15                        and                  16 = 6 + 10

8.      It is the only number that is the perimeter and area of the same square. This was proved by Pythagorean.                             

1	2	3	4
5	6	7	8
9	10	11	12
13	14	15	16

9.      It is the centered Pentagonal number.

10.  The atomic number of Sulphar is 16.

11.  The Chinese has developed a method of counting using the thumb to touch the tip and three joints of each finger. This enabled them to count to 16 on each hand.

12.  In web designing, colours are coded on base16 numbering system. The web coding changes the RGB values (0 – 255) into base 16.

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Importance of 15

                                            Fifteen

1.      It is a magic constant in a 3 x 3 magic square. This magic square was found on the back of tortoise by king Yo in China. This is the smallest non trivial magic square. An n x n magic square consists of the n² integers from 1 to n arranged in n x n square grid, such that the sum of any row, or any column, or of one of the two diagonals, is equal to any other.

2	9	4
7	5	3
6	1	8

2.      It is the fifth triangular number

 

                                            

3. It is the product of two odd primes.                                                          

                                               15 = 3 x 5

4.      It can be expressed as the sum of two consecutive numbers and interestingly it can be expressed as the difference of square of these two consecutive numbers.                           

           15 = 8 + 7

= 8² - 7²

5.      The numbers of 3 digit palindromic primes are 15. 

These are- 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 787, 797, 919 and 929

6.      The atomic number of Phosphorus is 15.

7.      It can be portioned into 176 ways.

8.      It is the name of famous slide puzzle where 15 numbers are arranged in a 4 x 4 square in zig  zag order and the      player is required to arrange these numbers.             

                                                                     

                                       

9.      Fifteen is a solo card game in which you lay out 16 cards at a time and try to clear the deck by removing combinations that adds up to 15.

10.  Fifteen squares can be arranged in a beautiful manner such that it produces 3 stars.                      

                           

11.  15 and 21 are the smallest pair of triangular numbers whose sum and difference are also triangular. The next such pair is 780 and 990.                                                                              

                                              15 + 21 = 36                                                          

                                              21 – 15 = 6                  

12. Fifteen is the 3rd Pentatope Number. It is a number in the 5th cell of any row of Pascal Triangle

13. 15 is the maximum number of pieces that can be produced from a cylindrical cake with four planar slices. A cake number is  maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. 

Watch the video of cutting cake at :- 

https://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Cake_number_with_4_cutting_planes.gif/220px-Cake_number_with_4_cutting_planes.gif

(Sources :- Wikipedia)

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Number 14

Fourteen (14)            

  

1.      It can be portioned into 135 ways.

2.     
It is the sum of square of first three natural numbers; therefore it is called the square pyramidal number.                                                                                                                            

  14 = 1² + 2² + 3²   

3.      It is the 7^th composite number.

4.      It is the product of two primes 2 and 7.

5.      It is a Catalan number. It is the only semi primes among all Catalan number. Catalan numbers are generated by the formula-   

C_n = 2n/ n! (n+1)!                                                                                              

6.      A figure having 14 faces with triangles and square is called Cuboctahedron. It has 6 squares and 8 equilateral triangles.

                                                                   

7.      It is the smallest even number n such that φ(m) = n has no solution. The sequence of such numbers goes to 14, 26, 34, 38, 50, 62, 68, 74 and so on.

8.      It is the atomic number of Silicon. The maximum number of electron that can fit in f sub level is 14.

9.      A fortnight has 14 days.

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