2007 UNCO Math Contest II Problems/Problem 6
Problem
(a) Demonstrate that every odd number 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 :
(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:
Therefore 2n is a multiple of 4.
Case 2: 2n is the difference of two odd numbers:
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 by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |