MathLabs

Problem 5

Let a,ba,b be positive integers. Dividing a2+b2a^2+b^2 by a+ba+b gives quotient qq and remainder rr. Find all pairs (a,b)(a,b) such that q2+r=1977q^2+r=1977.
Step 4 of 4: List and verify the pairs
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(a,b)∈{(7,50),(37,50),(50,7),(50,37)}(a,b)\in\{(7,50),(37,50),(50,7),(50,37)\}
Detailed analysis

The offsets from 2222 are ±15\pm15 and ±28\pm28. Positivity leaves exactly the four displayed ordered pairs. Substitution gives quotient 4444 and remainder 4141 in each case, so all four satisfy q2+r=1977q^2+r=1977.