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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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.
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.
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