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

Rajesh Kumar Thakur

rkthakur1974@gmail.com