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

Using Euclid division algorithm, we know that a=bq+r, 0≤r<b—(1) 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, or 0≤r<6. Therefore, a=6q or 6q+1 or 6q+2 or 6q+3 or 6q+4 or 6q+5
6q+0: 6 is divisible by 2, so it is an even number.
6q+1: 6 is divisible by 2, but 1 is not divisible by 2 so it is an odd number.
6q+2: 6 is divisible by 2, and 2 is divisible by 2 so it is an even number.
6q+3: 6 is divisible by 2, but 3 is not divisible by 2 so it is an odd number.
6q+4: 6 is divisible by 2, and 4 is divisible by 2 so it is an even number.
6q+5: 6 is divisible by 2, but 5 is not divisible by 2 so it is an odd number.
And therefore, any odd integer can be expressed in the form 6q+1 or 6q+3 or 6q+5