Question

Prove that the fraction is irreducible for every positive integers and .

Hard

Solution

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Let us consider three cases as follows:

Case 1: When is even and is even, then and .Β Therefore,

which is not reducible.

Case 2: When is even and is odd, then and .Β Therefore,

which is not reducible.

Case 3: When is odd and is odd, then and .Β Therefore,

which is not reducible.
Β 
Hence the fractionΒ  is irreducible for every positive integer and .

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