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GCD and Ordered pair | AIME I, 1995 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on GCD and Ordered pair.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on GCD and Ordered pair.

GCD and Ordered pair – AIME I, 1995


Find number of ordered pairs of positive integers (x,y) with \(y \lt x \leq 100\) are both \(\frac{x}{y}\) and \(\frac{x+1}{y+1}\) integers.

  • is 107
  • is 85
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

GCD

Ordered pair

Check the Answer


Answer: is 85.

AIME I, 1995, Question 8

Elementary Number Theory by David Burton

Try with Hints


here y|x and (y+1)|(x+1) \(\Rightarrow gcd(y,x)=y, gcd(y+1,x+1)=y+1\)

\(\Rightarrow gcd(y,x-y)=y, gcd(y+1,x-y)=y+1\)

\(\Rightarrow y,y+1|(x-y) and gcd (y,y+1)=1\)

\(\Rightarrow y(y+1)|(x-y)\)

here number of multiples of y(y+1) from 0 to 100-y \((x \leq 100)\) are

[\(\frac{100-y}{y(y+1)}\)]

\(\Rightarrow \displaystyle\sum_{y=1}^{99}[\frac{100-y}{y(y+1)}\)]=49+16+8+4+3+2+1+1+1=85.

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