Question

Show that any positive odd integer is of the form 4q+1 & 4q+3 where q is some integer ?

Answer 1

Step-by-step explanation:

Let a be any odd positive integer. we have to prove that a is of the form 4q + 1 or 4q + 3, where q is some integer.

Since a is an integer, consider b = 4 as another integer.

Applying Euclid's division lemma, we get:

a = 4q + r for some integer q ≥ 0 and r = 0, 1, 2 and 3, since 0 ≤ r < 4.

Therefore, a = 4q or 4q + 1 or 4q + 2 or 4q + 3.

However, since a is odd, it cannot take the values 4q or 4q + 2 (since all these are divisible by 2).

Hence, any odd integer can be expressed in the form 4q + 1 or 4q + 3, where q is some integer.

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