Question

Any positive odd integer is in the form of 4 q + 1 or 4q + 3 where q is some integer.
true or false

Answer 1

$\huge\bold \red{True }$

Explanation:

By Euclid’s division algorithm,

a = bq + r

Take b = 4

∴ Since 0 ≤ r < 4,r = 0, 1, 2, 3

So, a = 4q, 4q + 1, 4q + 2, 4q + 3

Clearly, a = 4q, 4q + 2 are even, as they are divisible by 2.

Therefore 'a' cannot be 4q, 4q + 2 as a is odd. But 4q + 1, 4q + 3 are odd, as they are not divisible by 2.

∴ Any positive odd integer is of the form (4q + 1) or (4q + 3).