100737

P and Q represent different positive integers.  Find all values of both P and Q such that the product of the product of P and Q and the difference of P less Q is equal to 100737.  In other words, .
Source: mathcontest.olemiss.edu 10/11/2010

Solution
First, let’s write the prime factorization of .
The problem asks us to rewrite as a product of three factors, i.e. .
Example 1:
Example 2:
In Example 1, if we let and , then , so this is not a solution.
In Example 2, if we let and , then , so this is not a solution.

Notice that the factors are always odd, because there is no in the prime factorization of .
In Example 1, and . In Example 2, and .

Thus, the third factor must be odd, but this is not possible because when you subtract two odd numbers you always get an even number.

We conclude that there are no different positive integers and such that .
We can check our answer by looking at the list of all possible groups of three factors below:

3	3	11193	100737
3	7	4797	100737
3	9	3731	100737
3	13	2583	100737
3	21	1599	100737
3	39	861	100737
3	41	819	100737
3	63	533	100737
3	91	369	100737
3	117	287	100737
3	123	273	100737
7	9	1599	100737
7	39	369	100737
7	117	123	100737
9	13	861	100737
9	21	533	100737
9	39	287	100737
9	41	273	100737
9	91	123	100737
13	21	369	100737
13	63	123	100737
21	39	123	100737
21	41	117	100737
39	41	63	100737