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April 27, 2020

Odd and Even integers | AIME I, 1997 | Question 1

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

Check the Answer


Answer: is 750.

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.

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