Question

Show that every positive even integer is of the form 2q, and that every
positive odd integer is of the form 2q + 1, where q is some integer.
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Answer 1

$\huge\mathfrak\blue{Solution :-}$

Let any positive(even/odd) integer be a. By Euclid's Lemma we know that,

a = 2q + r where, 0 ≤ r < 2

Here, r can be equal or greater than zero but less than 2 at any cost.

This possible values for r can be 0 or 1.

r = 0 and 1

By substituting values of r,

a = 2q and a = 2q + 1

$<i><b><u>$

Thus, a will be an even positive integer for 2q.

Similarily, a will be an odd positive integer for 2q + 1