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

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Best answer

Let a is any positive odd integer and b = 4.

Applying Euclid division lemma in a, b

Where o ≤ r < 4 and q is any integer.

r = 0, 1, 2, 3 put

a = 4q + 0 

⇒ a = 4q

a = 4q + 1

a = 4q + 2

a = 4q + 4

For positive odd integer.

a ≠ 4 q, a ≠ 4q + 2

Hence, any odd integer is of the form 4q + 1, or 4q + 3.

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