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

1) \(\frac{m}{3}\) is divisible by 3

2) \(\frac{m}{3}\) has two different prime factors

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.

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..
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"Nowadays, people know the price of everything, and the value of nothing."Oscar Wilde

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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Re: If m is divisible by 3, how many prime factors does m have? [#permalink]

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25 Mar 2013, 20:52

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!
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MV "Better to fight for something than live for nothing.” ― George S. Patton Jr

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.

Re: If m is divisible by 3, how many prime factors does m have? [#permalink]

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27 Mar 2013, 16:59

@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!?

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!?

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

1) \(\frac{m}{3}\) is divisible by 3

2) \(\frac{m}{3}\) has two different prime factors

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.

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
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Re: If m is divisible by 3, how many prime factors does m have? [#permalink]

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18 Feb 2014, 06:12

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

Re: If m is divisible by 3, how many prime factors does m have? [#permalink]

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15 Sep 2015, 23:41

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