e-GMAT
A positive integer n has the smallest 3 prime numbers as its only prime factors. How many positive integers divide n completely?
1) The total number of times the prime factors of n occur in n is 5.
2) The product of the number of times each prime factor of n occurs in n is 4.
OA C
A positive integer n has the smallest 3 prime numbers as its
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The smallest 3 prime numbers are 2, 3 and 5.AAPL wrote:e-GMAT
A positive integer n has the smallest 3 prime numbers as its only prime factors. How many positive integers divide n completely?
1) The total number of times the prime factors of n occur in n is 5.
2) The product of the number of times each prime factor of n occurs in n is 4.
OA C
Say n = 2^a * 3^b * 5^c, where a, b and c are positive integers
The number of factors of n are (a + 1) * (b + 1) * (c + 1). We have to get the value of (a + 1) * (b + 1) * (c + 1).
Let's take each statement one by one.
1) The total number of times the prime factors of n occur in n is 5.
=> a + b + c = 5
Case 1: Say a = 1, b = 2 and c = 2; then the number of factors of n = (a + 1) * (b + 1) * (c + 1) = (1 + 1) * (2 + 1) * (2 + 1) = 2*3*3 = 18.
Case 2: Say a = 1, b = 1 and c = 3; then the number of factors of n = (a + 1) * (b + 1) * (c + 1) = (1 + 1) * (1 + 1) * (3 + 1) = 2*2*4 = 16.
No unique value. Insufficient.
2) The product of the number of times each prime factor of n occurs in n is 4.
=> a*b*c = 4
Case 1: Say a = 1, b = 2 and c = 2; then the number of factors of n = (a + 1) * (b + 1) * (c + 1) = (1 + 1) * (2 + 1) * (2 + 1) = 2*3*3 = 18.
Case 2: Say a = 1, b = 1 and c = 4; then the number of factors of n = (a + 1) * (b + 1) * (c + 1) = (1 + 1) * (1 + 1) * (4 + 1) = 2*2*5 = 20.
No unique value. Insufficient.
(1) and (2) together
The only common value in the two statements is 18. Sufficient.
The correct answer: C
Hope this helps!
-Jay
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