## [ Decimal numeral system ]  [ Divisibility of numbers. General properties ]  $[$ Divisibility rules for 2 and 4 $]$  $[$ Divisibility rules for 3 and 9 $]$

The number $42 X 4 Y$ is divisible by 72. Find its digits $X$ and $Y$.

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Since 72 is divisible by 4, by the divisibility rule for 4, we get that $Y$ is also divisible by 4. Since 72 is also divisible by 9, by the divisibility rule for 9,

$X+Y+10$ is divisible by 9, which is only possible if $X+Y=8$ or $X+Y=17$. Combining these results and using the fact that $X, Y$ are digits, we get the possible pairs: $(X, Y)=(8,0),(4,4),(0,8),(9$, 8). Substituting all these pairs into the original number, we can verify that only two of them:

$(0,8)$ and $(8,0)-$ are valid.

## Answer

$X=0, Y=8$ or $X=8, Y=0$.