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1400
Ordered Pair Equation — Editorial
number theorydiophantine equationsfactorization
Rearranging x2+y=xy+12x^2 + y = xy + 12 gives y(x−1)=x2−12,y(x-1) = x^2 - 12, so
y=x+1−11x−1.y = x + 1 - \frac{11}{x-1}.
For yy to be a positive integer, x−1x - 1 must be a positive divisor of 11.11. Since 1111 is prime, x−1∈{1,11},x - 1 \in \{1, 11\}, giving x∈{2,12}.x \in \{2, 12\}. For x=2x = 2: y=−8y = -8 (not positive). For x=12x = 12: y=12y = 12 (positive). The only solution is (12,12),(12, 12), so there is 11 ordered pair.