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If m is divisible by 3, how many prime factors does m have? [#permalink]
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22 Jun 2010, 00:12
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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 I've seen answers given as B , C and E by different people. I would go for E 1. if m has 3 and m/3 has a 3, then m has at least 3^2, but m may have other factors.. 2.m has a 3 and 2 different primes (x,y) from m/3. So m has at least 3,x,y if not more. But either x or y could be a 3 as well. in that case m has 3^2,x or 3^2,y 1+2 m/3 has a 3 and another prime factor x m then has 3^2, x, and other factors that we dont know about According to mgmat book, since m is a variable, we couldn't tell how many prime factors if we don't have constraint m Could someone post the correct solution? OPEN DISCUSSION OF THIS QUESTION IS HERE: ifmisdivisibleby3howmanyprimefactorsdoesmhave126571.html
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Last edited by Bunuel on 22 Oct 2013, 23:53, edited 1 time in total.
Renamed the topic, edited the question and added the OA.



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Re: Prime factor:Can someone give a definite answer? [#permalink]
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22 Jun 2010, 03:06
Pipp wrote: According to mgmat book, since m is a variable, we couldn't tell how many prime factors if we don't have constraint mwhat does this mean ?
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Re: Prime factor:Can someone give a definite answer? [#permalink]
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22 Jun 2010, 03:11
m is divisible by 3. How many prime factors does m have?
1. Remember, we also consider zero to be perfectly divisible by any number, and hence m could be some multiple of 3 greater than 3, or zero. INSUFF
2. m/3 has 2 prime factors. say x and y. So m = 3 * x^k * y^n If m = 2*3*5, then m/3 = 2*5. So m has 3 prime factors If m = 3^2 * 5 then m/3 = 3*5, but m has only 2 prime factors, insufficient.
Both 1 and 2.
Since m/3 is divisible by 3, one of the prime factors of m/3 is 3 So m/3 = 3^n * y^k => m = 3^(n+1) * y^k Thus m also has 2 prime factors.
Pick C.
Remember we are only concerned with prime factors, which are counted only once each irrespective of how many times they occur in the prime factorization of a number.



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Re: If m is divisible by 3, how many prime factors does m have? [#permalink]
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22 Oct 2013, 16:11
Pipp wrote: If m is divisible by 3, how many prime factors does m have? 1). m/3 is divisible by 3. 2). m/3 has two different prime factors. I've seen answers given as by different people. I would go for E 1. if m has 3 and m/3 has a 3, then m has at least 3^2, but m may have other factors.. 2.m has a 3 and 2 different primes (x,y) from m/3. So m has at least 3,x,y if not more. But either x or y could be a 3 as well. in that case m has 3^2,x or 3^2,y 1+2 m/3 has a 3 and another prime factor x m then has 3^2, x, and other factors that we dont know about According to mgmat book, since m is a variable, we couldn't tell how many prime factors if we don't have constraint m Could someone post the correct solution? Hi Bunuel Can you explain this question Regards



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Re: If m is divisible by 3, how many prime factors does m have? [#permalink]
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22 Oct 2013, 23:55
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. OPEN DISCUSSION OF THIS QUESTION IS HERE: ifmisdivisibleby3howmanyprimefactorsdoesmhave126571.html
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Re: If m is divisible by 3, how many prime factors does m have?
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