If ($x * 10^q$) - ($y * 10^{r}$) = $10^r$, where q, r,x and y are positive integers and $q>r$, then what is the units digit of $y$?

Question

Q. If ($x * 10^q$) - ($y * 10^{r}$) = $10^r$, where q, r,x and y are positive integers and $q>r$, then what is the units digit of y?

My approach:

$10^r$ (x * $10^{(q-r)}$ - y ) = $10^r$

$x* 10^{(q-r)}$ = y

So, answer will be zero because q-r >0 and anything multiplied by $10^x $ will have a units digit =0.

But the answer is 1.

Answer 1

You simplified incorrectly: when you divide both sides by $10^r$, you should get $$x\cdot 10^{q-r}-y=\color{red}1\;,$$ so that $y=x\cdot 10^{q-r}-1$. And since $q-r>0$, $x\cdot 10^{q-r}$ ends in $0$, and the units digit of $y$ must be $9$, not $1$.