Question

show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5 where q is some integer

Answer 1

here is your answer hope it will help:)-

Answer 2

Euclid’s division algorithm, for two positive  integers a and b, we have

a = bq + r, 0 ≤ r < b

Let b = 6,

r = 0, 1, 2, 3, 4, 5

Hence

a = 6q, 6q + 1, 6q + 2, 6q + 3,

6q + 4, 6q + 5

Clearly, a = 6q, 6q + 2, 6q + 4 are even and divisible by 2.

But 6q + 1, 6q + 3, 6q + 5 are odd and not divisible by 2.

Any positive odd integer is of the form 6q + 1,  6q + 3 or 6q + 5.