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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