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

Dear Student

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.

Now substituting the value of r, we get,
If r = 0, then a = 6q
Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2.
A positive integer can be either even or odd
Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is someinteger.

Thanks