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Difficulty:
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Question Stats:
66% (01:15) correct
34%
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based on 91
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If positive integer n is the product of 4 different prime numbers, including 1 and n, how many factors does n have?
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
I am getting 16
rule: add 1 to each exponent of each prime number, and multiply them together.
(1+1)(1+1)(1+1)(1+1) = 16
rule: add 1 to each exponent of each prime number, and multiply them together.
(1+1)(1+1)(1+1)(1+1) = 16
jallenmorris
Joined: 30 Apr 2008
Last visit: 09 Oct 2014
Posts: 1,226
Location: Oklahoma City
Schools:Hard Knocks
Posts: 1,226
good rule, mind telling us why we do it in addition to what we should do ?
gmatnub
IanStewart
Expert
GMAT Tutor
Joined: 24 Jun 2008
Last visit: 13 Sep 2026
Posts: 4,146
Location: United States
GMAT 1: 780 Q51 V47
Expert reply
GMAT 1: 780 Q51 V47
Posts: 4,146
jallenmorris
Suppose p, q and r are different primes, and p^3 * q^4 * r^2 is the prime factorization of n. How many positive divisors should n have?
Well, any number that can be written as p^a * q^b * r^c will be a divisor of n as long as:
a = 0, 1, 2 or 3 (four choices)
b = 0, 1, 2, 3 or 4 (five choices)
c = 0, 1 or 2 (three choices)
How many different sets of exponents a, b and c could we choose? This is a straight counting problem: we multiply the choices we have for each. Thus, n will have 4*5*3 = 60 divisors.
This all works because of unique factorization into primes, which guarantees that each of the divisors we get above will be different, and that n has no other divisors besides the 60 we just found.
Short version: we add one to each power because in a divisor, the power on any prime can be zero. We multiply because essentially it's just like any other counting problem.
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ianstewartgmat.com
ianstewartgmat.com
If positive integer n is the product of 4 different prime numbers, including 1 and n, how many factors does n have?
N =a*b*c*d, where a,b,c,d are different prime numbers
Therefore, no. of factors = (1+1)(1+1)(1+1)(1+1)= 16
Hence C
N =a*b*c*d, where a,b,c,d are different prime numbers
Therefore, no. of factors = (1+1)(1+1)(1+1)(1+1)= 16
Hence C
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