Vincen wrote:If n has 15 positive divisors, inclusive of 1 and n, then which of the following could be the number of divisors of 3n?
I. 20
II. 30
III. 40
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II and III only
We see that n has an odd number of divisors (or factors), so it must be a perfect square, since only perfect squares have an odd number of factors. Also recall that in order to obtain the number of factors of a number, we add one to each of the exponents of the primes in the prime factorization of the number and multiply the results. For example, 24 = 2^3 x 3^1, and thus 24 has (3 + 1) x (1 + 1) = 8 factors.
Since we are given that n has 15 factors, n must be one of the following formats (note: p and q are primes):
1) n = p^14 (we see that n has 14 + 1 = 15 factors)
2) n = p^2 x q^4 (we see that n has (2 + 1) x (4 + 1) = 15 factors)
We are asked for the number of factors of 3n. Let's analyze each case above:
Case 1: n = p^14
If p = 3, then 3n = 3 x 3^14 = 3^15, and hence 3n has 15 + 1 = 16 factors.
If p ≠3, then 3n = 3^1 x p^14, and hence 3n has (1 + 1) x (14 + 1) = 30 factors.
Case 2: n = p^2 x q^4
If p = 3, then 3n = 3 x 3^2 x q^4 = 3^3 x q^4, and hence 3n has (3 + 1) x (4 + 1) = 20 factors.
If q = 3, then 3n = 3 x p^2 x 3^4 = 3^5 x p^2, and hence 3n has (5 + 1) x (2 + 1) = 18 factors.
If p ≠3 and q ≠3, then 3n = 3^1 x p^2 x q^4, and hence 3n has (1 + 1) x (2 + 1) x (4 + 1) = 30 factors.
Of the three given Roman numerals, we see that only I and II are possible.
Answer:
B
