How did we know that X must be equal to 1?

Question

I did not understand the solution to this problem

Answer 1

Consider the expression $$ Y + 10 Z = 89 X $$ where $X, Y$ and $X$ are all single non-zero digits, or they can only be $1, 2, 3, 4, 5, 6, 7, 8$ or $9$. What can the value of $X$ be? We know that $Y+10Z$ can at it maximum be $9 + 90 = 99$, so certainly the right-hand side cannot be more than $99$. Therefore the only sensible value for $X$ is $X=1$.

Answer 2

It is already explained after $Y+10Z=89X$. The left hand side maximum value is $98$. The right hand side is a multiple of $89$, so it can be $89$, $2\cdot89=178$, $3\cdot89=267$,... Obviously the only good value is $89$, which corresponds for $X=1$