Question

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

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Answer 1

Let be any positive integer

We know by Euclid's algorithm, if a and b are two positive integers, there exist unique integers q and r satisfying, where.

Take

Since 0 ≤ r < 4, the possible remainders are 0, 1, 2 and 3.

That is, can be , where q is the quotient.

Since is odd, cannot be 4q or 4q + 2 as they are both divisible by 2.

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

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