Find the numbers

Question

The sum of two positive integers is $200$. If one is divided by $5$ and the other is divided by $9$, the remainder is $1$ each case. Find the numbers

I have $u+v=200$. But I can't get the second equation.

Answer 1

One must be of the form $5r+1$ and the other of the form $9s+1$. Thus you wish to solve $5r+1+9s+1=200$ or $5r+9s=198$. The solution is not unique unless you add more constraints. For example the numbers $181$ and $19$ add to $200$ and have the property that you can divide $181$ by $5$ and $19$ by $9$ and get remainders of $1.$ The numbers $126$ and $64$ also have this property.

Answer 2

Given the conditions on $u$ and $v$, we can find $m$ and $n$ such that \begin{align*} u &= 5m + 1 \\ v &= 9n + 1. \end{align*} Substituting in, we get $$5m + 9n = 198$$ From here, we get a few answers. For example, $n = 2$ and $m = 36$, yielding $u = 181$ and $v = 19$. See if you can find some others.