MathLabs

Problem 4

Determine all pairs (a,b)(a,b) of positive integers such that ab2+b+7 ab^{2}+b+7 divides a2b+a+b a^{2}b+a+b .
Step 1 of 6: Apply the division algorithm
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In plain words

Eliminate the highest-degree terms.

ab2+b+7∣7a−b2ab^2+b+7\mid 7a-b^2
Detailed analysis

Let D=ab2+b+7 D=ab^2+b+7 and E=a2b+a+b E=a^2b+a+b . Since aD−bE=7a−b2 aD-bE=7a-b^2, we have D∣7a−b2 D\mid 7a-b^2.