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## Number Properties - Playing Devil's Advocate with Prime Factors

Confess it – while watching Harvey Specter and Mike Ross on ‘Suits’, many of you have wondered how ‘cool’ it would be to be a lawyer. It’s surprising how they question every assumption, every reason and come up with an innovative solution which looks as if the magician just pulled out a rabbit out of a hat.

Well, in high level questions, you have a chance to play the Devil’s Advocate. If your best thought out logic says that answer has to be 2, still think why it cannot be 1. The higher level questions are quite tricky and if you play at 700+ level, you will need to be extra careful – if it seems too easy, it probably is! To illustrate, we have quite a brilliant little question from GMAT Prep.

Question: If 5x^2 has two different prime factors, at most how many different prime factors does x have?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Solution: So here is the logic with which most of us would come up – 5x^2 has two different prime factors – one would be 5 since it is already there so x must have one more prime factor. x^2  has only those prime factors that x has so 5x^2 will have two prime factors - one 5 and the other from x. Sounds perfectly reasonable and the answer should be 1 – x has 1 prime factor. In fact, it must have at least one prime factor and it cannot have more than one prime factor.

But then, and here we have a hint in the question to play the devil’s advocate – why does the question ask “at most how many different prime factors”. If there were a single value for different prime factors of x, the question would have probably said “how many different prime factors…”. There would have been no need for those words ‘at most’.

Then look at the other options. Is it possible that x has 2 prime factors? It certainly cannot have more than 2 distinct prime factors since then, 5x^2 will have more than two distinct prime factors. Actually, x can have two prime factors! x can have 5 as a factor too. Sneaky – eh? We already have a 5 in 5x^2 but that doesn’t mean that we cannot have a 5 in x^2 too. It will still count as a single prime factor. x can have another prime factor such as 2 or 3 or 7 or 11 etc. In that case, x can have two distinct prime factors.

So x can be 15 (two different prime factors) and x can be 25 (one prime factor)

Note that this question has no calculations and no time consuming equations but still, this little trick makes this question quite hard. If most people get it wrong because of missing this trick, the question will be termed hard.

Now here is a trickier version of this question:

Question 2: How many prime factors does positive integer n have?

Statement 1: n/7 has only one prime factor.
Statement 2: 3*n^2 has two different prime factors.

Solution: Let’s keep in mind our learning from above while trying to solve this question.

Statement 1: n/7 has only one prime factor.

n/7 has a factor so obviously, it is an integer. Hence n must have a 7 as a factor. So we might jump to the conclusion that n has two prime factors –7 and another one which is left when n is divided off by 7. So n would be something like 7*3 so that n/7 = 7*3/7 = 3 (only one prime factor).

But what we wouldn’t have considered in this case is that n may have multiple 7s so that when a 7 is cancelled in n/7, you would still be left with 7 i.e. if n is 7*7, then n/7 = 7*7/7 = 7. In this case, n has only one distinct prime factor.

So n can have either one or two prime factors. This statement alone is not sufficient.

Statement 2: 3*n^2 has two different prime factors.

This is the same as our previous question. 3n^2 has two different prime factors but n itself can have either one or two prime factors (one of which will be 3). For example, n can be 7 or n can be 3*7. This statement alone is not sufficient.

Using both statements, n could have one or two prime factors i.e. n could be 49 (only one prime factor - 7) or n could be 21 (two prime factors).

Hence, even using both the statements, we cannot say how many prime factors n has.