Difference of squares
Number Theory
Level
5
How many positive integers less than 1000 CANNOT be expressed as the difference of two perfect squares?
The answer is 250.
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1 solution
Remember that: a 2 − b 2 = ( a + b ) × ( a − b )
So any number which can be expressed as the difference of two perfect squares should be either a product of two odd numbers, or a product of two even numbers.
Therefore, any number which CANNOT be expressed as the difference of two perfect squares must necessarily be an odd multiple of 2.
There are 250 odd multiples of 2 less than 1000.