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Prove that the square of an odd number is always 1 more than a multiple of 4

Let n be any whole number. Any odd number can be written as 2n+1. Any odd number squared is therefore (2n+2)2=2n2n+22n+1=4n2+4n+1=4(n2+n)+1. n2+n is a whole number, so 4(n2+1) is a multiple of 4. Therefore, any odd number squared is 1 more than a multiple of 4.

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