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

Question

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

RandomStoryteller · Accepted Answer

All we got to remember:
LCM of Fractions = LCM(Numerator)/ GCF(Denominator)
GCF pf Fractions = GCF (Denominator) / LCM (Numerator).
Solution in attached image.

Kinshook · Answer

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

Bunuel · Answer

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

cs15811581 · Answer

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

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