Show that any positive odd integral number can be expressed as 6q + 1 or 6q + 3 or 6q + 5 where q is a positive integer.

Let take a as any positive integer and b = 6.
Then using Euclid’s algorithm we get a = 6q + r here r is remainder and value of q is more
than or equal to 0 and r = 0, 1, 2, 3, 4, 5 because 0 ≤r < b and the value of b is 6

So total possible forms will 6q + 0 , 6q + 1 , 6q + 2,6q + 3,6q + 4,6q + 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 also 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 also divisible by 2 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

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5