Question

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

Answer 1

Let a is b be two positive integers in which a is greater than b.

According to Euclid's division algorithm; a and b can be expressed as a = bq + r, where q is quotient and r is remainder and 0 ≤ r < b.

Taking b = 4, we get: a = 4q + r, where 0 ≤ r < 4 i.e., r = 0, 1, 2 or 3

r = 0 ⇒ a = 4q, which is divisible by 2 and so is even.

r = 1 ⇒ a = 4q + 1, which is not divisible by 2 and so is odd.

r = 2 ⇒ q = 4q + 2, which is divisible by 2 and so is even. and

r = 3 Þ q = 4q + 3, which is not divisible by 2 and so is odd.

Any positive odd integer is of the form

4q + 1 or 4q + 3; where q is an integer.

Hence the required result.