GMAT CLUB OLYMPICS: What is the number of factors of a positive intege

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.


Answer: D

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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