Is n dvisible by at least 6 positive integers?

[mehravikas]

S11-22 If n is a positive integer, is n divisible by at least six positive integers?

(1) n is the product of three different prime numbers.
(2) n = 30

[netigen]

Ans is D

[mehravikas]

Can you explain with an example please?

From my understanding, statement 1 is insufficient. Let n = 5 i.e. 5 * 3 * 2, in this case n is not divisible by 6 positive integers.

[Ian Stewart]

Can you explain with an example please?

From my understanding, statement 1 is insufficient. Let n = 5 i.e. 5 * 3 * 2, in this case n is not divisible by 6 positive integers.

If n = 2*3*5, then n is divisible by 1, 2, 3, 5, 6 (=2*3), 10 (=2*5), 15 (=3*5) and 30 (=2*3*5) - eight different positive integers. In fact, if n is equal to the product of three different primes p, q and r, then n = pqr, and n must be divisible by:

1, p, q, r, pq, pr, qr, and pqr

All of these numbers must be different, because numbers have unique prime factorizations. Every number that has three distinct prime divisors has at least eight positive divisors. If n = pqr, where p, q and r are different primes, then n has precisely 8 divisors.

So each Statement is sufficient (the second statement is a special case of the first). D.

[mehravikas]

Thanks. I was thinking that divisible by 6 integers means exactly 6.

Can you explain with an example please?

From my understanding, statement 1 is insufficient. Let n = 5 i.e. 5 * 3 * 2, in this case n is not divisible by 6 positive integers.

If n = 2*3*5, then n is divisible by 1, 2, 3, 5, 6 (=2*3), 10 (=2*5), 15 (=3*5) and 30 (=2*3*5) - eight different positive integers. In fact, if n is equal to the product of three different primes p, q and r, then n = pqr, and n must be divisible by:

1, p, q, r, pq, pr, qr, and pqr

All of these numbers must be different, because numbers have unique prime factorizations. Every number that has three distinct prime divisors has at least eight positive divisors. If n = pqr, where p, q and r are different primes, then n has precisely 8 divisors.

So each Statement is sufficient (the second statement is a special case of the first). D.