# Show that any prime number can be expressed in the form 6k+_1(k belongs to natural no)for no. greater than 3

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by sushant21

2015-04-05T01:34:56+05:30

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5 = 6 *1 - 1        7 = 6 * 1 + 1        11 = 6 * 1 - 1
13 = 6 * 1 + 1        17 = 6 * 3 - 1          19 = 6 * 3 + 1
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All integers can be written in the form as :

6 k,  6k + 1 , 6 k + 2 , 6 k + 3, 6 k + 4, 6 k + 5

Each integer from -∞ to ∞  falls in one of the above six forms.  In the above:
6k is divisible by 6.          6k+2 is divisible by 2.        6 k + 3 is divisible by 3.
6 k + 4 is divisible by 4.      6k + 5 is same as 6 (k+1) - 1  in the same form as 6 k -1.

Hence, all prime numbers can be expressed in the form:  6k +1 or  6k -1.