If m is divisible by 3, how many prime factors does m have?

If m is divisible by 3, how many prime factors does m have?

(1) \(\frac{m}{3}\) is divisible by 3 --> \(\frac{m}{3}=3k\) --> \(m=3^2*k\) --> \(m\) has at least one prime 3, but it can have more than one, in case \(k\) has some number of other primes. Not sufficient

2) \(\frac{m}{3}\) has two different prime factors --> first of all 3 is a factor of \(m\), so 3 is one of the primes of \(m\) for sure.

Now, if power of 3 in \(m\) is more than or equal to 2 then \(m\) will have have only two prime factors: 3 and one other, example: \(m=18\), (as in \(\frac{m}{3}\) one 3 will be reduced, at least one more 3 will be left, plus one other, to make the # of different factors of \(\frac{m}{3}\) equal to two. Thus \(m\) will have 3 and some other prime as a prime factors).

But if \(m\) has 3 in power of one then \(m\) will have 3 prime factors: 3 and two others, example \(m=30\) (one 3 will be reduced in \(m\) and \(\frac{m}{3}\) will have some other two prime factors, which naturally will be the primes of \(m\) as well). Not sufficient.

(1)+(2) From (1) \(3^2\) is a factor of \(m\), thus from (2) \(m\) has only two distinct prime factors: 3 and one other. Sufficient.

Answer: C.
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Just one thought bunuel..... The question asks for "how many prime factors". It does not ask for "how many DIFFERENT prime factors". If combined the statements tell us that \(3^2\) is a prime factor and there is a different prime factor as well, we still do not know how many times that "different" prime factor repeats. So my question is, when the question ask for number of prime factors, does it mean we should be looking at "different" prime factors or should we count the different factors each time they are repeated as well..

Well, the real GMAT question will ask about "distinct primes", to avoid such technicalities, though we can say that it's implied here.
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Hi Bunuel,

I answer B, since statement 2 says it has 2 different primes factor (3 included for sure) answering directly to the question. Why (2) it's insufficient? I didn't understand your approach, please explain. Thanks!

The second statement says that m/3 has two different prime factors, NOT m.

If m = 18, then m/3 = 6 (6 has two different prime factors: 2, and 3). 18 has two different prime factors 2 and 3.
If m = 30, then m/3 = 10 (10 has two different prime factors: 2, and 5). 30 has three different prime factors 2, 3 and 5.

Thus this statement is NOT sufficient.

Does this make sense?
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Given:

m= 3s


Statement 1:

m = 9r

Not sufficient as r may be formed of 1 or more prime factors.


Statement 2:

m/3 = p*q

Not sufficient because p or q may be 3.

So we need to find whether p or q is 3.


Taken together:

1. Since m is divisible by 9 from statement 1, m/3 is divisible by 3.

2. From (1) above , we can say that the RHS in statement 2 is a product of 3. i.e., one of p or q is 3.

3. From (2) above, we can say that m has two different prime factors since m/3 = p*q and if say p is 3, then m=3*3*q

Therefore the answer is choice C.
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Yes Bunuel, thanks!

Posted from my mobile device

@Bunuel

second statement says m has two different prime factors. Can it be taken as it has only 2 different prime factors? if m/3 is 42, the second statement is still true!?

Please check here: if-m-is-divisible-by-3-how-many-prime-factors-does-m-have-126571.html#p1202604
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Hi Bunnel,

Can you please explain me the statement.

(1)+(2) From (1) \(3^2\) is a factor of \(m\), thus from (2) \(m\) has only two distinct prime factors: 3 and one other. Sufficient.

I didn't get how you got the answer after combing the two

My way:

First of all we are told that m = 3k where 'k' is an integer. How many different prime factors does 'm' have?

Statement 1: m = 9k, nothing about prime factors
Statement 2: m/3 has two different prime factors. Well if m = 3k then 3k/3 = k has two different prime factors but no info on 'm' yet. 'k' could have 3 among its factors or not. Therefore since we are asked about different prime factors then this statement alone is not sufficient.

Statements 1 and 2 together tell us that 9k/3=3k has two different prime factors. Since m = 3k then it must be that 'k' is another different prime factors Therefore C is our answer

Hope this helps
Cheers
J

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