Unwholesome

Prove that if n is a positive integer greater than 1, then

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$

is not an integer.

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Find the least common multiple of 2, 3, …, n, and recast each fraction using this denominator. Now all the numerators will be even except for one term, the one whose original denominator had been the highest power of 2 that doesn’t exceed n. Thus the sum of the numerators is odd while the common denominator is even.