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Question

# Let x,y∈Z such that x2−2x=y2−2y+1010. Then the number of pairs (x,y) satisfying the equation is

A
only one
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B
infinitely many
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C
more than one but finite
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D
no such pair is possible
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Solution

## The correct option is D no such pair is possiblex2−2x=y2−2y+1010 ⇒x2−2x+1=y2−2y+1+1010 ⇒(x−1)2=(y−1)2+1010 Let x−1=X and y−1=Y Then, X2−Y2=1010 ⇒(X−Y)(X+Y)=1010 Let both X and Y be even. Then (X−Y)(X+Y) is a multiple of 4. But 1010 is not a multiple of 4. If X is even and Y is odd or vice-versa, then (X−Y)(X+Y) is odd. If both X and Y is odd, then (X−Y)(X+Y) is a multiple of 4. So, no value of x and y is possible.

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