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(a^2+b^2)/(a-b)^2 is an integer => (a^3+b^3)/(a-b)^3 is an integer

Source: 2022 Turkey JBMO TST P1

March 15, 2022

number theoryIntegersDivisibility

Problem Statement

For positive integers

a

and

b

, if the expression

\frac{a^2+b^2}{(a-b)^2}

is an integer, prove that the expression

\frac{a^3+b^3}{(a-b)^3}

is an integer as well.