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

Solution:
Let a be any positive integer and b = 6.
Then, by Euclid’s algorithm, a = 6q + r for
some integer q ≥ 0 and r = 0,1,2,3,4,5 because
0 ≤ r ≤ 6.
So, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
Here, a cannot be 6q or 6q + 2 or 6q + 4, as they are divisible by 2.
6q + 1
6 is divisible by 2 but 1 is not divisible by 2.
6q + 3
6 is divisible by 2 but 3 is not divisible by 2.
6q + 5
6 is divisible by 2 but 5 is not divisible by 2.
Since, 6q + 1, 6q + 3, 6q + 5 are not divisible by 2, they are odd numbers.
Therefore, any odd integer is of the form
6q + 1, or 6q + 3, or 6q + 5.