#### Question

If m, n\in Z^+ such that m+n+mn=34, find the value of m+n

If m, n\in Z^+ such that m+n+mn=34, find the value of m+n

Express m in terms of n from the given condition

\because m +n+mn=34

m(1+n)=34-n

m=\dfrac{34-n}{1+n}

=\dfrac{35-n-1}{1+n}

=\dfrac{35}{1+n}-1

In order for m+n to be integers, 35 must be divisible by n+1 . In other words n+1 is equal to one of factors of 35. So we get

n+1 = 5 or n+1 = 7

m+n = 6 or m+n =4