Can you find the number?

Question

There's a number with the following characteristics:

• The hundreds digit plus the units digit minus the tens digit equals 8.
• 3 times the hundreds digit plus 2 times the tens digit minus the units digit equals 33.
• The number divided by the sum of its digits equals 53.

What's the number?

Answer 1

The number is

$954$,

because we can rewrite the three given conditions as follows, where $H$, $T$, and $U$ are the hundreds, tens, and units digits respectively:

$H+U-T=8,\quad 3H+2T-U=33,\quad 100H+10T+U=53H+53T+53U$

Adding the first two gives

$4H+T=41$, therefore $T=41-4H$. Since both $T$ and $H$ must be numbers from $0$ to $9$, the only possibilities are $H=8,T=9$ and $H=9,T=5$.

Also from the first equation,

$U=8+T-H=8+(41-4H)-H=49-5H$. Here the only possibilities are $H=8,U=9$ and $H=9,U=4$.

So the number must be

either $899$ or $954$. Only one of these is a multiple of $53$, namely $954=53\times18=53\times(9+5+4)$, so the problem is solved.