Problem Solving

goyalsau wrote:How many real n's are there such that n! is a perfect square?

(E) More than 3
(C) 2
(D) 3
(B) 1
(A) 0

fskilnik asked for my input on this. It's a deeply flawed question for several reasons; anyone studying for the GMAT can simply ignore it and stop reading this post now, since nothing below will be relevant to you. But since I wouldn't want to describe a practice question as 'flawed' without explaining why, I'll mention three different reasons why this is a terrible GMAT practice question:

* the question asks about *real* numbers 'n' for which n! is a perfect square. Well, in advanced math, you learn how to define n! even when n is a decimal number (the function you get is a translation of what is known as the 'gamma function'). That function is continuous and constantly increasing, so can equal absolutely any perfect square greater than 0. So a mathematician would say that the answer to this question is 'infinitely many'. If the OA to this question is '2', the OA is simply wrong. The math involved is, however, beyond even what you'd encounter in a standard undergraduate math curriculum, and is simply light years beyond what is required on the GMAT.

* here, there are certainly at least two small integers n for which n! is a square; n = 0 and n = 1. That said, I have never seen a real GMAT question which tests whether you know what 0! means. It's the kind of technicality the GMAT question designers seem to go to great lengths to avoid testing, and I expect there never will be a real GMAT question which requires you to know anything about 0!. Since this is one of the 'traps' built into this question, it's simply not realistic.

* further, if you assume n is an integer, then to answer this question properly, you need to prove that there is *no* large integer value of n for which n! is a square. That's impossible to do using only what a GMAT test taker would normally know, since it requires you to use some kind of information about the distribution of primes. The conventional way to prove this is to use something known as Bertrand's Postulate (which is not a postulate, but a proven theorem), as fsklinik did above. That postulate is extremely difficult to prove from scratch (there's absolutely no way anyone could prove it within two minutes and apply it to this question), and it will never, ever, be required to solve a real GMAT question. So while you might encounter this question if you take a Masters level course in Number Theory, it is millions and millions of miles too advanced for the GMAT.

Any prep material that presents this as a realistic GMAT practice problem should simply be thrown in the rubbish bin, since if the authors think the GMAT tests things like this, they simply don't understand what's on the test. Glancing at the other questions in the original post, they all suffer from the same issue. Where are they from?