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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Number 13

Thirteen (13)       

1.      Thirteen is the second smallest prime p, the period of whose reciprocal is ½ (p – 1 ). In simple word any numbers from 1/13 to 12/13 have periods 6.                                                            

  1/13 = 0.076923076923---                                                                                                  2/13=  0.153846153846---                                                                                                      3/13 = 0.230769230769---                                                                                                          ------------------------------                                                                                                 12/13 = 0.923076923076---

2.      The smallest solution for the Pellian equation x² – ny² = 1 for which x = 649 and y = 180 is n = 13.

3.      It can be portioned in 101 ways.

4.      It is the only integer that can be expressed as the sum of two consecutive squares. Not only that its fourth power is the sum of two consecutive squares.                                                 

      13 = 2² + 3²                                                                                                                             13⁴ = 119² +120²

5. There are 13 Archimedian solids named after Archimedes.

6.      It is the hypotenuse of a right angled triangle with two of its side 5 and 12.

7. It can be expressed as the sum and difference of two consecutive numbers.                                    13 = 7² – 6²                                                                                                                            13 = 2² + 3²      

8. If we subtract the sum of digits from 13, a perfect square is obtained.                                       13 – (1 + 3) = 9 = 3²

9.      If we add the product of its digits in 13, we get a perfect square.                                                    13 + (1 x 3 ) = 16 = 4²

10.      Any three digit number written twice in the numerical pattern of ABCABC is divisible by 13.                                                                                                                                                   234234 ÷ 13 = 18018                                         678678 ÷ 13 = 52206

11 .      The fear of 13 is called TRISKAIDEKAPHOBIA. This is therefore considered an unlucky number and many high rise building doesn’t have the floor number 13.    

12.      There are 13 cards of spades (♠), hearts ( ♥), diamonds (♦) and clubs (♣) in a well shuffled pack of playing cards.

             

            13.     P² – Q² is divisible by 13 only when P and Q are not divisible by 13. 

   

            14.      In a pine cones there are 13 spirals going anticlockwise and 8 in clockwise direction.                      The red line shows the 13 anticlockwise spirals in a pine cone

15.      It is a prime number so is its reverse. Moreover the square of 13 and its reverse 31 have the digits written in reverse order.                                                                                             

    13² = 169                           31² = 961

16.   Divisibility by 13: - To test whether a number is divisible by 13, remove the last digit, then add 4 times that digit to the remaining number. Repeat this process until you get the number 40 or less. If this process results in 0, 13, 26 or 39 the original number is divisible by 13.                                                                                                           

                Example: - Check whether 678678 is divisible by 13?                                                                   67867 + 4 x 8 = 67899                                                                                                    6789 + 4 x 9 = 6825                             

                                                                          682 + 4 x 5 = 702                                                                                                           70 + 4 x 2 = 78 is divisible by 13.     

                         

   17.  The moon moves 13 degree around the earth every day. Moreover, it takes 13 days to change from Full Moon to New moon and 13 days to change back with one full day and one day New to equal 28 days of Lunar cycle.

18.   It is a sixth prime number and a Fibonacci number also.

19.      It is one of the 3 known Wilson Prrimes. 

20.   It is the smallest prime number that can grow 6 times by the right. 

See the pattern below-                   

  13 is a prime                                                                                                                

   139 is a prime                                                                                                                       

   1399 is a prime                                                                                                                        13999 is a prime                                                                                                                  

139991 is a prime                                                                                                             

  1399913 is a prime                                                                                                         

13999133 is a prime

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Number 12

Importance of Twelve (12)

1.      Its etymology originates from the German compound twalf meaning two left over.

2.      It can be portioned in 77 ways.

3.      There are twelve months in a year. They are – January, February, March, April, May, June, July, August, September, October, November, and December.

4.      There are 12 zodiac signs. They are – Aries (March 21 – April 19) , Taurus ( April 20 – May 20) , Gemini (May 21 –June 20), Cancer (June 21- July 22) , Leo ( July 23 -  August 22 ), Virgo (August 23 – September 22) , Libra (September 23 – October 22) , Scirpio (October 23 - November 21), Sagittarius (November 22 - December 21), Capricorn (December 22 – January 19), Aquarius (January 23 - February 18), Pisces (February 19 – March 20)

5.      A 12 sided polygon is called a dodecagon and a 12 faced polyhedron is called a dodecahedron. A dodecahedron is a solid with regular pentagon on each of its 12 faces.

                                                                  

6.      Octahedron and Cube have 12 edges.

7.      The strength of wind is measured on the Beaufort scale which runs from 0 to 12.

8.      On one hand you can count off the numbers from 1 to 12 by touching the joints of your fingers with your thumb.                                                                        

                                         

9.      12 is the first abundant number. An abundant number is a number whose sum of factors excluding itself is more than the number. 12 has the factors as 1, 2, 3, 4, 6 and 12 and the sum of 1 + 2+ 3 + 4 + 6 =16 >12.

10.  It is a pentagonal number. Those numbers which represent the shape of pentagon are called pentagonal number. In the pentagonal numbers the lower base is a square with a triangle on the top. 1, 5,12,22,35...are its example. The nth pentagonal number P_n is given by the formula:-

a.       P_n = n ( 3n – 1 )

2

  

11.  It is the smallest number with exactly 6 divisors. Its divisors are 1, 2, 3, 4, 6 and 12.

12.  It is the largest known even number expressible as the sum of two primes in one way.                    12 = 5 + 7                               

13.  The square of 12 is palindromic to its reverse 21.                                                     

                    12² = 144                                                21² = 441

14.  It is the first composite number of the form p²q; a square prime combination.                            12 = 2² x 3

15.  It is a sublime number. A number that has a perfect number of divisors and the sum of its divisors is also a perfect number. 12 has 6 divisors (1, 2, 3, 4, 6 and 12) and 6 is a perfect number. Moreover the sum of its divisor is 28 which is also a perfect number.

16.  It is a super factorial number because it can be expressed as the product of first three factorials.                                                                                   

                                                  12 = 1! x 2! x 3!

17.  The atomic number of magnesium is 12. 

18.  The 12^th moon of Jupiter is Lysithea.

19.  The famous kissing number problem in mathematics talks about impossibility of arranging more than 12 spheres of equal size so as to touch a central sphere.

20.  There are many mathematical terms associated with 12.                                                           

   1 feet = 12 inch                           1 dozen  =12 articles

21.  It has a very interesting property.                                                                               1122334455667789 x 12² = 16 16 16 16 16 16 16 16 16     

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