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805+ Level|   Multiples and Factors|         
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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 20 Aug 2021, 08:00
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95% (hard)

Question Stats:

38% (01:58) correct 62% (01:47) wrong based on 247 sessions

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What is the number of factors of a positive integer n ?

(1) The least common multiple of integers \(\frac{n}{2}\) and \(\frac{n}{3}\) is 210
(2) The greatest common factors of integers \(\frac{n}{5}\) and \(\frac{n}{7}\) is 6


M36-20

 


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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 20 Aug 2021, 08:50
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All we got to remember:
LCM of Fractions = LCM(Numerator)/ GCF(Denominator)
GCF pf Fractions = GCF (Denominator) / LCM (Numerator).
Solution in attached image.
Attachments

1629474443220.jpg
1629474443220.jpg [ 2.58 MiB | Viewed 4541 times ]

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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 20 Aug 2021, 08:17
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Asked: What is the number of factors of a positive integer n ?

(1) The least common multiple of integers \(\frac{n}{2}\) and \(\frac{n}{3}\) is 210
Least common multiple of n/2 & n/3 = n = 210 = 2*3*5*7
Number of factors of positive integer n = 2*2*2*2 = 16
SUFFICIENT

(2) The greatest common factors of integers \(\frac{n}{5}\) and \(\frac{n}{7}\) is 6
Greatest common factor of integers n/5 & n/7 = n/35 = 6
n = 210 = 2*3*5*7
Number of factors of positive integer n = 2*2*2*2 = 16
SUFFICIENT

IMO D
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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 04 Oct 2021, 08:31
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Bunuel
What is the number of factors of a positive integer n ?

(1) The least common multiple of integers \(\frac{n}{2}\) and \(\frac{n}{3}\) is 210
(2) The greatest common factors of integers \(\frac{n}{5}\) and \(\frac{n}{7}\) is 6


M36-20

 


This question was provided by GMAT Club
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Official Solution:


What is the number of factors of a positive integer \(n\) ?

(1) The least common multiple of integers \(\frac{n}{2}\) and \(\frac{n}{3}\) is 210

First of all: \(210=2*3*5*7\).

Next:

\(n\) must be a multiple of 2 because we are told that \(\frac{n}{2}\) is an integer;

\(n\) must be a multiple of 3 because we are told that \(\frac{n}{3}\) is an integer;

\(n\) must be a multiple of 5 because either \(\frac{n}{2}\) or \(\frac{n}{3}\) is a multiple of 5 (how else 5 would appear in the LCM?);

\(n\) must be a multiple of 7 because either \(\frac{n}{2}\) or \(\frac{n}{3}\) is a multiple of 7 (how else 7 would appear in the LCM?);

\(n\) cannot have higher powers of 2, 3, 5, or 7 because in this case the LCM would also contain 2, 3, 5, or 7 in higher power;

\(n\) cannot be a multiple of any other prime because in this case that prime would also appear in \(\frac{n}{2}\) or \(\frac{n}{3}\) and thus in the LCM.

Thus, \(n\) must be \(210=2*3*5*7\). Sufficient.

(2) The greatest common factors of integers \(\frac{n}{5}\) and \(\frac{n}{7}\) is 6

\(n\) must be a multiple of 2 and 3. Because \(6=2*3\) is a factor of \(\frac{n}{5}\) and \(\frac{n}{7}\), then \(6=2*3\) must also be a factor of \(n\). In addition notice that \(n\) cannot have higher powers of 2 or 3 because in this case the GCF would contain higher powers of these primes;

\(n\) must be a multiple of 5 because we are told that \(\frac{n}{5}\) is an integer. \(n\) cannot have higher power of 5 because in this case the GCF would contain 5;

\(n\) must be a multiple of 7 because we are told that \(\frac{n}{7}\) is an integer. \(n\) cannot have higher power of 7 because in this case the GCF would contain 7;

\(n\) cannot be a multiple of any other prime because in this case that prime would also appear in \(\frac{n}{5}\) and \(\frac{n}{7}\) and thus in the GCF.

Thus, \(n\) must be \(210=2*3*5*7\). Sufficient.


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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 20 Aug 2021, 09:09
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What is the number of factors of a positive integer n ?
We have to find the value of n or power of prime factors.

(1) The least common multiple of integers \(\frac{n}{2}\) and \(\frac{n}{3}\) is 210
LCM of two fractions = LCM of numerator/HCF of denominator =\(\frac{LCM(n,n)}{HCF(2,3)}=\frac{n}{1}=n=210\)
Sufficient

(2) The greatest common factors of integers \(\frac{n}{5}\) and \(\frac{n}{7}\) is 6
HCF of two fractions = HCF of numerator/LCM of denominator =\(\frac{HCF(n,n)}{LCM(5,7)}=\frac{n}{35}=6……n=210\)
Sufficient


D
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GMAT CLUB OLYMPICS: What is the number of factors of a positive intege 28 Dec 2023, 02:37
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