A question on divisibility

Question

For what values of $x,y \in \{1,2,3,...9 \}$, does $$10x+y \space\mid 100x + y $$ ?

What approach should I take for solving this problem ?

Answer 1

You can just list them-there are $81$ cases to try if $x$ and $y$ are allowed to be equal, $72$ otherwise. This is not as silly as it sounds. Even if there are too many cases to list them all, trying some small ones sometimes is enlightening. Or you could use the fact that $a|b \leftrightarrow a|(b-a)$

Answer 2

Let $\dfrac{100x+y}{10x+y}=d\iff \dfrac xy=\dfrac{d-1}{100-10d}$

$\dfrac{100x+y}{10x+y}-10=-\dfrac{9y}{10x+y}<0\implies d<10\implies d\le9$

As $1\le x,y\le9,\dfrac19\le\dfrac xy\le9\implies\dfrac19\le\dfrac{d-1}{100-10d}\le9$

$\dfrac{d-1}{100-10d}\le9\iff d\le\dfrac{901}{91}<10\implies d\le9$

$\dfrac19\le\dfrac{d-1}{100-10d}\iff d\ge\dfrac{101}{19}>5$

$\implies6\le d\le9$

If $d=9,\dfrac xy=\cdots=\dfrac45,\dfrac x4=\dfrac y5=k$(say)$\implies y=5k$ as $1\le y\le9\implies y=5$

Try with $d=6,7,8$