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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Odd and Even integers.

## Odd and Even Integers – AIME I, 1997

Find the number of integers between 1 and 1000 that can be expressed as the difference of squares of two non-negative integers.

• is 107
• is 750
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

Divisibility

Difference of squares

But try the problem first…

Source

AIME I, 1997, Question 1

Elementary Number Theory by David Burton

## Try with Hints

First hint

Let x be a non-negetive integer $(x+1)^{2}-x^{2}=2x+1$

Second Hint

Let y be a non-negetive integer $(y+1)^{2}-(y-1)^{2}=4y$

Final Step

Numbers 2(mod 4) cannot be obtained as difference of squares then number of such numbers =500+250=750.