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PS Question

Expert replies
by adthedaddy » Thu Sep 24, 2015 7:10 pm
If n=3x4xp, where p is a prime number greater than 3, how many different positive non-prime divisors does n have, excluding 1 and n ?
A) Six
B) Seven
C) Eight
D) Nine
E) Ten
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Source: — Problem Solving |

by Matt@VeritasPrep » Fri Sep 25, 2015 12:34 am
Your number has a prime factorization of 3¹ 4² p¹, so it has (1+1) * (2+1) * (1+1) = 12 unique factors.

It has THREE unique primes (2, 3, p) and TWO invalid factors (1, n), so 12 - 3 - 2 = 7.

This is easily checked by choosing a p value, e.g. p = 5. 3 * 4 * 5 = 60, which has the factors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The factors we want are 4, 6, 10, 12, 15, 20, and 30, so there are 7.
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