# 2007 UNCO Math Contest II Problems/Problem 6

## Problem

(a) Demonstrate that every odd number $2n+1$ can be expressed as a difference of two squares.

(b) Demonstrate which even numbers can be expressed as a difference of two squares.

## Solution

(a) For any odd number $2n+1$: $2n+1 = n^2 + 2n +1 - n^2$ $2n+1 = (n+1)^2 - n^2$

(b) When expressing an even number as a difference of two numbers, these must be of the same parity, i.e. both must be either even or odd. The parity of a number stays the same when squaring it. This gives us two cases:

Case 1: 2n is the difference of two even numbers: $2n = (2m)^2 - (2p)^2$ $2n = 4(m^2 - p^2)$

Therefore 2n is a multiple of 4.

Case 2: 2n is the difference of two odd numbers: $2n = (2m+1)^2 - (2p+1)^2$ $2n = 4m^2 + 4m + 1 - 4p^2 - 4p - 1$ $2n = 4(m(m+1) - p(p+1))$

Therefore 2n is a multiple of 4.

For both cases we find that only even numbers which are multiples of 4 can be expressed as a difference of two squares.

## See Also

 2007 UNCO Math Contest II (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 All UNCO Math Contest Problems and Solutions
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