The Count.

Number Theory Level 2

How many ordered pairs of integers ( x , y ) (x, y) satisfy x + y = 100 x + y = 100 and gcd ⁡ ( x , y ) = 8 \gcd(x, y) = 8 ?


The answer is 0.

Mike Argus by Mike Argus , 21, USA

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2 solutions

Mike Argus
Aug 11, 2015

Proof: We will show that no solution exists. Assume there exists an ordered pair (x, y) such that x + y = 100 and gcd(x, y) = 8. Then 8 | x and 8 | y. So, 8 | x + y = 100, a contradiction. This completes the proof.

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Good observation that gcd ⁡ ( x , y ) ∣ x + y \gcd(x,y) \mid x+y .

Department 8
Aug 12, 2015

According to the question g c d ( x , y ) = 8 gcd(x,y) = 8 either x = 8 y x=8y or y = 8 x y=8x By seeing in case we see x = 100 9 x=\frac{100}{9} but it was given that x , y x, y are integers therefore no pairs are possible

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

This solution is currently incorrect. Can you spot the error?

Are you sure? 16 and 24 satisfy gcd ⁡ ( 16 , 24 ) = 8 \gcd(16, 24) = 8 , but we do not have 16 = 8 × 24 16 = 8 \times 24 or 24 = 8 × 16 24 = 8 \times 16 .

Calvin Lin Staff - 5 years, 10 months ago

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