Let ‘a’ be any odd positive integer we need to prove that a is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

Since a is an integer consider b = 6 another integer applying Euclid's division lemma we get

a = 6q + r for some integer q ≠ 0, and r = 0, 1, 2, 3, 4, 5 since 0 < r < 6.

Therefore according to question:

a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

However since a is odd so ‘a’ cannot take the values 6q, 6q+2 and 6q+4

(since all these are even integers, hence divisible by 2)

Also, 6q + 1 = 2 x 3q + 1 = 2A₁ + 1, where A₁ is a positive integer

6q + 3 = (6q + 2) + 1 = 2 (3q + 1) + 1 = 2A₂ + 1, where A₂ is a positive integer

6q + 5 = (6q + 4) + 1 = 2 (3q + 2) + 1 = 2A₃ + 1, where A₃ is a positive integer

Clearly, 6q + 1, 6q + 3, 6q + 5 are of the form 2A + 1, where A is a positive integer.

Therefore, 6q + 1, 6q + 3, 6q + 5 are odd numbers.