Importance of Number 7

         Characteristics of Seven (7)

1.  It is a prime number. Prime number is a number which has only two factors 1 and the number itself.

2.  There are seven days in a week. They are – Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday.

3. The oldest puzzle recorded in Ahmes Papyrus has another version like this—             

           As I was going to St. Ives,                                                                         

            I met a men with 7 wives.                                                                                                                Each wife had 7 sacks.                                                                                                               

        Each sack has 7 cats.                                                                                                        

        Each cat had 7 kits.                                                                                         

      Kits, cats, sack and wives                                                                     

  how many were going to St. Ives?                                                                      

This puzzle requires the knowledge of Geometric Progression. In simple words, the total number = 1 + 7 + 49 + 343 + 2401 = 2801.

4. Seven (7) is the congruent number because it is the area of the triangle with sides 35/12, 24/5 and 337/60. A triangle is said to be congruent if its area comes out to be an integer.

5. It is a Mersenne Prime number. A prime which can be written as 2ⁿ – 1 is said to be Mersenne Prime.                                                                                                                                                            7 = 2³ – 1

6. The sevenths power of number 14459929 has a very interesting feature.  7^th  Power of every digit of 14459929 sums up to 14459929.                                                                           

           14459929 = 1⁷ + 4⁷ + 4⁷ + 5⁷ + 9⁷ + 9⁷ + 2⁷ + 9⁷

7. The factorial of seven (7!) is 5040. This result contains 4 digits with 2 zeros, i.e. 50% zero. Probably it is the unique factorial with this property.

8. No perfect square ends with 7.

9. The smallest positive integer whose reciprocal 1/7 has a pattern of more than one repeating digit.  Example:- ¹/₇ = 0.142857 142857 142857….This number 142857 shows an explicit feature when it is multiplied by 1, 2, 3, 4, 5 and 6. Let’s see it-                                                                    

  142857 x 1 = 142857                                                                                                 

    142857 x 2 = 285714                                                                                                       

   142857 x 3 = 428571                                                                                                        

   142857 x 4 = 571428                                                                                                     

   142857 x 5 = 714285                                                                                                                      142857 x 6 = 857142                                                                           

Did you notice anything? Yes, the digit of the product comes in a cyclic order and when you prepare a multiplicative table on the basis of the result of the above product the sum of each row and column comes out to be 27.

Digit	1	4	2	8	5	7	Digit Sum
1	1	4	2	8	5	7	27
2	2	8	5	7	1	4	27
3	4	2	8	5	7	1	27
4	5	7	1	4	2	8	27
5	7	1	4	2	8	5	27
6	8	5	7	1	4	2	27
Column sum	27	27	27	27	27	27

10. Seven can be portioned in 11 ways.

11. The seven bridges of Konigsberg in Germany gave birth to new branch of Mathematics called Topology. Leonard Euler (1707-1783) solved this problem.

                                        

12. No perfect square ends with 7. Any number that ends with 2, 3, 7 or 8 can’t be a perfect square.

13. It is a Lucas Number. 1, 3, 4, 7, 11, 18… are the Lucas number. In Lucas Number, the next number is the sum of previous two.

14. It is the sum of 1^st and 3^rd factorials. 1! + 3!  = 7

15. It is the first centered hexagonal number.

                                       

1.      There are 7 different ways of linking four hexagons together. One of them is shown here.

                

2.       The opposite side of a die cube sum up to 7.

1.      It is the first prime which can be written in the form of 6m + 1, where m is a positive integers.

2.       Here is an interesting game based on 7.                                                                                 

               7 is a Prime Number                                                                              

             73 is a Prime Number                                                                                               

            739 is a Prime Number.                                                                                            

           7393 is a Prime Number.                                                                                           

         73939 is a Prime Number.                                                                                                     

        739391 is a Prime Number.                                                                                                   

      7393913 is a Prime Number.

3.      In the Fermat’s Last Theorem xⁿ + yⁿ = zⁿ, for n = 7 this equation has no solution was proved by Lame in 1840.

14. The sum of the squares of the first seven primes was 666, the supposed number of the beast from Revelations.
666=2^2+3^2+5^2+7^2+11^2+13^2+17^2

15. The smallest prime Palindrome number 7778777 has only two digits in it ,i.e 7 and 8.

  

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        Rajesh Kumar Thakur

      rkthakur1974@gmail.com

Number 6

                           Important Feature  of 6

1.      It is the second smallest composite number its proper divisor are 1, 2 and 3. A number is composite if it has more than two factors.

2.      There are six (6) basic trigonometric function- Sine, Cos (ine), Tan(gent), Cot (angent), Sec (ant) , Cosec (ant). These ratios define the ratio of two sides of a right angled triangle.

3.       Six (6) is the only even prime number that is not the sum of successive odd cubes because six is the sum of first three natural numbers.    
                                                                     6 = 1 + 2 + 3

4.      Bee hive is hexagonal in shape.

                                                                                           

5.      A benzene molecule has a ring of 6 carbon atom.                 

            

6.      Atomic number of Carbon is six.

7.      It is the first number which is divisible by  1, 2, 3                                                                              6 = 1 + 2 + 3

8.      Six is the only number whose sum of factor and product comes out to 6.                                         6 = 1 + 2 + 3             and 6 = 1 x 2 x 3

9.      It is the first Perfect Number. Perfect number is a number whose sum of factors excluding itself is equal to the number itself.                                                                                                                    6 = 1 + 2 + 3                                                                                                      
 Saint Augustin writes that although God could have created the world all at once, he preferred to take six days as 6 is symbolized by the Perfect Number.  

10.  It can be portioned in 11 ways.

11.  Six also has a special geometrical relationship with circles. If you take six identical circular coins and place them around other coins of the same size so that they touch it, they will all touch each other too whatever the size of the coin you take.

12.   It is a third triangular number.

13.  A cube has 6 faces it is also known as hexahedron.

14.  It is the highest number on a normal die. A die has number 1 to 6 written on it.

               

15.   Six (6) is a congruent number because it is the area of a right angle triangle having side 3, 4 and 5. A congruent number is an integer that is the area of a right triangle with three rational number side.

16.  The Pythagoreans associated 6 with marriage and health, because it is the product of their first even and first odd numbers, which were female and male respectively. 

17.  It is the only number which is the sum and product of same 3 numbers.                                           6 = 1 + 2 + 3  and 6 = 1 x 2 x 3

18.  Any number of the form of 6m – 1 has two factors whose sum is divisible by 6, where m is any integer.

19. Every prime greater than three is either one more, or one less than a multiple of six.

20. There are five equable (area and perimeter "equal" and integer side lengths) triangles . All of them have area/perimeter divisible by six.

Example, a right angled triangle with sides 5, 12 and 13 has area 30 square unit and perimeter 30 units.

There are infinitely many Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10). There are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).
[It was proved in 1904 by W. A. Whitworth and D. Biddle]





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Rajesh Kumar Thakur
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