If n has 15 positive divisors. . . .

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

The OA is B.

How can I make this PS question? Should I set a particular value for n or should I make it for a general n? I am stuck here. Experts please help me.

Since n has 15 positive divisors

Number of divisors = (p+1)(q+1)(r+1)... where p, q, r etc are the powers of prime numbers of n

15 may be (2+1)*(4+1) or it may be (14+1)

I. 20 = (3+1)*(4+1)may be the factors of 3n if n had two powers of 3 earlier and 4 powers of some other prime number
II. 30 = (1+1)(2+1)*(4+1)may be the factors of 3n if n had no powers of 3 earlier, 2 powers of one prime number other than 3 and 4 powers of some other prime number
III. 40 = (7+1)*(4+1) CAN NOT be the factors of 3n cause theere is only one increase in power that can be expected after n is multiplied by 3

Answwer: Option B

To find the number of divisors of a number, find its prime factorization, count the number of each distinct prime, add one to each count, then multiply the counts together.

For instance, the process for 60:

1) Find its prime factorization: 60 = 2 * 2 * 3 * 5

2) Count each distinct prime: TWO 2s, ONE 3, ONE 5

3) Add one to each count: (TWO + 1), (ONE + 1), (ONE + 1)

4) Multiply these together: 3 * 2 * 2 => 12

If our number has fifteen unique factors, we must have multiplied (2 + 1) * (4 + 1), so we've got two of one prime and four of another: p * p * q * q * q * q, where p and q are distinct primes.

If p = 3, then 3n means we're adding another q: p * p * p * q * q * q * q, which would have (3 + 1) * (4 + 1) => 20 factors.

If q = 3, then 3n means we're adding another q: p * p * q * q * q * q * q, which would have (2 + 1) * (5 + 1) => 18 factors

If neither p nor q is 3, then we've got 3 * p * p * q * q * q * q, which would have (1 + 1) * (2 + 1) * (4 + 1) => 30 factors

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