Show that any positive odd integer is of the form (4q + 1) or (4q + 3), where q is a positive integer.

Question

Philoid · Accepted Answer

Let a be an positive odd integer, and let b = 4

By, using Euclid's division lemma,

a = 4q + r, where r is an integer such that, 0 ≤ r < 4

So, only four cases are possible

a = 4q or

a = 4q + 1 or

a = 4q + 2 or

a = 4q + 3

But 4q and 4q + 2 are divisible by 2, therefore these cases are not possible, as a is an odd integer.

Therefore,

a = 4q + 1 or a = 4q + 3.