



 
Andrew, if I understand you properly n.p is n p and you are looking at (np)^{p}/(c+np) and wants to know about the relation of c to np whenever this ratio is an integer. If this ratio is an integer, every prime q that divides the denominator must divide the numerator. Thus q is p or a divisor of n. If q is in fact p then we must have that p divides c + np which means that p divides c so that c is not relatively prime to np. If q is not p then it divides n and also divides the denominator c + np, but this means that q divides c so that c is not relatively prime to np. One way to look at the converse (or the above if you like) is to write it as (np)^{p} = k(c+np) for some integer k. Now any divisor of the right hand side must divide the left hand side but if c is relatively prime to np then c+np has a prime divisor q that doesn't divide n or p and thus q cannot divide the left hand side, a contradiction. Penny  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 