Let n>1 be an integer. Show that 1+1/2+1/3+...+1/n can never be an integer.?

Question

Im not sure if you have to use induction or not, but I am pretty sure divisibility is involved.

Answer 1

Okay, add up all the fractions, which will get you:

(1/n!)( (2*3...n) + (1*3...n) + ....+ (1*2...(n-2)(n)) + (1*2...(n-1)) )

Notice that each term in the numerator is missing one number in the series 1 to n. Consider the largest prime number p < n, so that p appears only once in the series 1 to n. All but one of the terms will have that prime number. That means p in the denominator n! will divide into all of those terms except that one which is missing that p. Hence, the sum cannot be an integer.

Answer 2

Someone else just answered this exact question. Look a page or two down in the mathematics section.