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

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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Rajesh Kumar Thakur

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Number 11

Eleven (11)

1.      Its etymology originates from a German compound ainlif meaning one left.

2.      Eleven is the only palindromic prime with an even number of digits.

3.      It can be portioned in 56 ways

4.       It is a Lucas Number. The numbers in Lucas series are 1, 3, 4, 7, 11…

5.      It is the fourth number that stays the same when written upside down. The first three are 0, 1 and 8. The others such numbers less than 100 are 69, 88, 96.

6.      The sun spot cycle repeats every 11 years. Sun spots are dark areas on the surface of the sun which have strong magnetic field.

7.Any power of 11 ends with 1.                                                                                                                       11² =121                                  11³ =1331  

     

8.The magical sum of this magic triangle is 11.                    

                                                                 

9.      It is the smallest three digits prime in ternary system and four digit prime in binary system. 11 in ternary system is written as (102)₃ and in binary as (1011)₂.

10.  It is the fourth Sophie Germain Prime. A prime number p is a Sophie Germain prime if 2p+ 1 is also a prime. The first ten Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83 and 89.

11.   It is a Stormer number. A Stormer number is a positive integer n for which the greatest prime factor of n² + 1 meets or exceeds 2n. It is named after Carl Stormer. The first eight Stormer numbers are 1, 2, 4, 5, 6, 9, 10, and 11.

12.  When you divide a number with 11 in denominator it will have two digit repeating sequences in their decimal expansion.                                                                                            

    1/11 = 0.090909..                                                    2/11 = 0.181818..                                

     3/11 = 0.272727...                                                 4/11 = 0.363636..

13.  Divisibility by 11:- There are several methods to check the divisibility of a number by 11 but the most common amongst them is to alternately add and subtract digits starting from the right. If the result thus obtained is the multiple of 11 then the original number will be divisible by 11.                                                                                                             Example:-Is 1331 divisible by 11?                                                                                           Here, 1 – 3 + 3 – 1 = 0, hence it is divisible by 11.

Rajesh Kumar Thakur

rkthakur1974@gmail.com

Number 10

                                            Number 10

       1.      It is the base of our numerical system.

       2.      It is the fifth even number. The other four preceding 10 are 2, 4, 6 and 8.

       3.      It is the fourth triangular number. The first three are 1, 3 and 6.

      4.      It is the sum of square of two consecutive odd numbers.                

                                                  10 = 1² + 3²

      5.      October 10, 2010 is written as 10/10/2010. The occurrence of such number is special because the first tow number added together equals the first two digits of the year. Here 10 + 10 = 20 is the first two digit of the year 2010. Such date occurs in every 201 years. The next such combination will occur on 11/11/2211.

        6.      It is the fifth composite number. The first four are 4, 6, 8 and 9.

       7.      Divisibility by 10:- If a number ends with zero it is exactly divisible by 10.         

                          Example:- 100, 560540

       8.      Roman numerals for ten is X. It represents two crossed hands.

     9.      Take any triangle and divide each side into three equal parts. Draw lines from these to the opposite corners. A new shape will appear in the middle of the triangle whose area is 1/10^th of the area of whole triangle. This is known as Marion Walter Theorem. This theorem state - the area of the central hexagonal region determined by trisection of each side of a triangle and connecting the corresponding points with the opposite vertex is given by 1/10 the area of the original triangle.

10.  The strength of earth quake is measured on the Richter scale. Each point on the Richter scale represents a tenfold increase in energy. The Richter scale is a logarithmic scale used to express the total amount of energy released by an earthquake. Its values typically fall between 0 and 9, with each increase of 1 representing a 10 fold increase in energy.      

Value	1	2	3	4	5	6	7	8	9	10
Potential Hazard	Extra-ordinary	Outstanding	Far reaching	High	Note worthy	Intermediate	Moderate	Minim	Low	insignificant

     11.  It is the sum of first three primes.                                                                                                

       10 = 2 + 3 + 5

    12.  It is the sum of three consecutive (3^rd, 4^th and 5^th) Fibonacci numbers.

   13.  The only known solution to                                                                                                            

             n! = a! b!                                                                 is  10! = 6! X 7!

There are only 11 powers of 10 that are products of two integers without any zero. These are--                                                                                                                                           

           10⁰ = 1                         10¹ = 2 x 5                            10² = 4 x 25                             

            10³ = 8 x 125               10⁴ = 16 x 625                         10⁵ = 32 x 3125                                                 10⁶ = 64 x 15625         10⁷ = 128 x 78125                   10⁹ = 512 x 1953125             

14. This beautiful formula according to Ramanujan came into his dream

15. 10! seconds = 6 weeks

16. Ten is the smallest semi-prime that is the sum of all the distinct prime numbers from its lower factor through its higher factor. 
10 = 2 x 5 
10 = 2 + 3 + 5

17. It is the sum of first four factorials
0! + 1 ! + 2 ! + 3 ! = 10 

18. It is a tetrahedral number or triangular pyramidal  number.
1 4,  10, 20 

      Read more on Mathematical Number in my book THE POWER OF MATHEMATICAL NUMBER available on website 

Rajesh Thakur

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Importance of Number 9

                              Mathematical  Characteristics of Nine

      It is the least number that can be expressed as the sum of cubes of first two numbers.              

                                  1³ + 2³ = 9

2.      It is the largest one digit number in Hindu- Arabic Number system.

3.      Casting Out Nines is the popular method to check the four fundamental operations ( + , _ , x and ÷)

4.      If the sum of the digits of a number is the multiple of nine it is divisible by nine.              

                    58527 is divisible by 9 as 5 + 8 + 5 + 2 + 7 = 27 is divisible by 9.

5.      If a number from 1 to 8 is divided by 9 the number which is being divided will be repeated infinitely after decimal.                                                                                  

                1/9 =0.1111….                    2/9 =0.2222….                     3/9 =0.3333….                                                    4/9 =0.4444….                    5/9 =0.5555….                    6/9 =0.6666….                                                                7/9 =0.7777….                    8/9 =0.8888….              

6.      It is a composite number with its proper divisor being 1 and 3.

7.      A polygon with nine sides is called a nonagon.

8.      If you multiply any number by 9 the sum of all digits of the product adds to 9.                                                   578329 × 9 = 5204961 and 5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9   

               482729235601 × 9 = 4344563120409 and  

                                            4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9

9.      The difference between a base 10 positive integer and the sum of its digits is a multiple of nine.                                                                                                                                               

     Example:- Let us take the number 768. The sum of its digit is 7 + 6 + 8 = 21. The difference between 768 and 21 is 768 – 21 = 747 which is divisible by 9.

10.  Nine is the atomic number of Fluorine.

11.  In Geometry, a circle that passes through the nine points of a triangle is known as Nine point circle. These nine points are- the midpoint of each side of triangle, the foot of each altitude and the midpoint of the line segment from each vertex of the triangle to the ortho center. The nine point circle is also known as Euler’s circle or Feuerbach’s circle.                        

12.  The first 9 numbers can be arranged in a magic square so that all rows, columns and both diagonals have the same sum 15.

4	9	2
3	5	7
8	1	6

13.  Nine is the maximum number of cubes that one needed to sum to any positive integer.

14.  Take any 3 digit number, invert it and subtract the smallest from the larger. The middle digit will always be 9.                                                                                                               

                        542 – 245  =  297                                                                                                                       

                       861 – 168 = 693

15.  It is the sum of factorials of 3 consecutive numbers.                                                       

                  1! + 2! + 3! = 1 + 2 + 6 = 9

16.  It is the fourth composite number. A composite number has more than 2 factors. 

                    9 has the factors 1, 3 and 9.

17.  It is the sum of first 3 odd numbers.                                                                                                  

                    1 + 3 + 5 = 9

18.  The nine cubes on the size dish stamp make an impossible figure. This arrangement can’t exist in real world.  This is a Swedish postal stamp.

19.  The following pattern shows the use of 0 to 9 used once.                                                

             



       9          =  



97524 / 10836
                                                                                                                                          
= 95823 / 10647
                                                                                                                                           
= 95742 / 10638
                                                                                                                                       
= 75249 /08361
                                                                                                                                         
= 58239 / 06471
                                                                                                                                      
= 57429 / 06381

20.  Let us play an interesting puzzle on multiplying of a number by 9.                                                   Multiply -- 10,112,359,550,561,797,752,808,988,764,044,943,820,224,719 by 9 and you will get the result by simply moving the last nine up to the front. Hence the answer is 910,112,359,550,561,797,752,808,988,764,044,943,820,224,71