You can put this solution on YOUR website!
When it comes to prime factorization, there are very few shortcuts and one is generally stuck using an algorithm to find all the factors.
For this problem, one thing you could do is look at and look at the remainder when dividing by the various prime numbers:
7^7 mod 43 = 7 ==> 43 (= 7 + 36) divides evenly into (7^7 + 36)
Once you know that, however, you need to divide 823543 by 43 to get 19153. Then you must continue to check this for divisibility by prime numbers (up to or 138.39) to try to factor this part.
19153/2 no (clearly!)
19153/3 no
19153/5 no
19153/7 no
19153/11 no
...
19153/107 yes! its 179 (179 is also prime so the task is complete at this step).
So we found 43, 107, and 179 divide evenly into 7^7+36.
