Find all integers $a,b$ such that $\frac{b^{b} +b}{a\cdot b^2 +9}$ is an integer.

number-theory

Solution 1:

Here is a partial solution:

  • $\forall{n\in\mathbb{N}}:a=0,b=18n$
  • $\forall{n\in\mathbb{N}}:a=0,b=18n+8$
  • $\forall{n\in\mathbb{N}}:a=0,b=18n+9$
  • $\forall{n\in\mathbb{N}}:a=9,b=36n+27$

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