Strayer Elementary Number Theory 1. 5. 72a.

If you hit a roadblock on this problem, then feel glad - you probably did it right. If you deduced that a+b=max{a,b}+min{a,b} is involved, you probably did it right.

It's just that, for m+n=p^(min{a,b})(p^(max{a,b}-b)q+p^(max{a,b}-a)r), we don't have the theory (if it even exists) to determine what values of p (if any) divide p^(max{a,b}-b)q+p^(max{a,b}-a)r. Not sure if the author realized this or not when tasking this problem.

For example, 3^(1)||6 and 3^(2)||18 and 3^(min{1,2})=3^(1)||6+18=3(2+6)=24. So min{a,b} is the correct answer in this instance, because we can't factor any positive powers of 3 out of 2+6=8. But on the other hand, 3^(1)||15 and 3^(1)||12 and 3^(3)||15+12=3(5+4)=27. So min{a,b} is not the correct answer because we can factor an additional positive power of 3 out of 5+4=9. In this instance min{a,b}+d, where d is the highest power of 3 dividing 5+4=9, would be the correct answer.

So take my solution with a grain of salt. Not sure if it would hold up under the scrutiny of a professor (I've been out of school a while). The critical part of me says my answer is essentially min{a+b}+d=anything. Telling your professor "anything" is the answer to this problem will probably get you in trouble. So it may be safe to say which cases that min{a,b} is not the correct answer, and then show for which cases min{a,b} is the correct answer. Godspeed.