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2021-08-12, 06:53   #2
axn

Jun 2003

23·233 Posts

Quote:
 Originally Posted by bur A sum will never have a prime factor that isn't a prime factor of all addends. Since for all primes <= pn there is a addend that doesn't contain it, the sum, i.e. the numerator, will not be divisible by any of p1 ... pn.
This is not quite correct. The important point is that pi divides _all_ terms except for _one_. If more than one term was not divisible, you can't say anything about divisibility of the whole sum. But in this case, we know that everything else is divisible by pi, and therefore the whole is _not_ divisible.

Rest of the argument is fine, I think. You can't simplify that fraction any further (since none of the prime factors of the denominator appears in the numerator, by the above argument).