Pair of positive integers in product sums

Question

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not answer.

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$

I just thought about this question and wanted to know how would you solve something like this. Is there a formula that we can use? Is there a certain way we can do this? Out of curiosity, what would the solution look like? Can someone please show me how to do this?

I know the factors of $2014$ are $2*19*53$ and the factors of $2015$ are $5*13*31.$ I know that $mn+m+n$ $=$ $(m+1)(n+1).$ How would I figure out the integers for $m$ and $n$ though.

Answer 1

Hint: If $mn+m+n = 2014$, then $2015 = mn+m+n+1 = (m+1)(n+1)$.

So $m+1$ and $n+1$ are complementary factors of $2015 = 5 \cdot 13 \cdot 31$.

Can you list all the factors of $2015$?

Answer 2

You (and Jimmy) have done the hard work, now just finish it off.

You know the factors of 2015, and you know $(m+1)(n+1) = 2015$

All you need to do now is rearrange the equation as follows:

$n+1 = \frac{2015}{m+1}$

$n = \frac{2015}{m+1} -1$

And substitute the factors of 2015 into $(m+1)$ to find $n$.

Eg. $n = \frac{2015}{m+1} -1,\ where\: (m+1) = 5$

$n = \frac{2015}{5} -1$

$n = 402,\ m=4$

Repeat for the other factors where $m<n$.