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

Answer:

Let a be any positive integer and b = 2. Then, by Euclid’s algorithm,

a = 2q + r

for some integer q ≥ 0, and r = 0 or r = 1, because 0 ≤ r < 2. So, a = 2q or 2q + 1.

If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 2q + 1.

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