Prove that if a positive integer is of the form 6q+5, then it is of the form 3q+2 for some integer q, but not conversely.

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## Rahul

## Arjun

Let a = 6q + 5β a = 6q + 3 + 2β a = 3(2q + 1) + 2β a = 3m + 2 where, m = 2q + 1So, If a positive integer is of the form (6q + 5), it can be written in the form of (3q + 2).Now,Let a = 3q + 2 for some integer q β₯ 0A positive integer is either of the form 2k or (2k + 1).When q = 2ka = 3(2k) + 2 = 6k + 2 which is not in the form of (6q + 5).When q = 2k + 1a = 3(2k + 1) + 2 = 6k + 5 which is in the form of (6q + 5).Hence, if a positive integer is of the form (3q + 2), it may or may not be in the form of (6q + 5).