Re: HOT Competition: If n is a positive integer, then n is a multiple of
[#permalink]
26 Aug 2020, 16:49
If n is a positive integer, then n is a multiple of how many positive integers?
(1) n5∗n(−2)n5∗n(−2) has exactly 5 divisors between 1 and itself, both not inclusive.
(2) 3n23n2 has exactly 8 divisors between 1 and itself, both not inclusive
Statement 1. SUFFICIENT
n^3 has 5 divisors not including 1 and n^3
prove values for n
if n is a prime number "p", the divisors are: 1 , p, p^2, p^3 -> only 2 divisors that are ok with the statement, we need 5 divisors
if n is a product of two equal prime numbers "p", the divisors are: 1 , p, p^2, p^3, p^4, p^5, p^6 -> 5 divisors. This is ok with the statement.
if n is a product of two different prime numbers "p" and "q", the divisors are: 1 , p, p^2, p^3, q, q^2, q^3, pq,.... -> more than 5 divisors. we need 5 divisors.
n is a multiple of 3 positive integers
Statement 2. SUFFICIENT
3 x n^2 has 8 divisors not including 1 and 3 x n^2
prove values for n
if n is a prime number "p", the divisors are: 1 , 3, p, p^2, 3p, 3p^2 -> only 4 divisors that are ok with the statement, we need 8 divisors
if n is a product of two equal prime numbers "p", the divisors are: 1 , p, p^2, p^3, p^4, 3, 3p, 3p^2, 3p^3, 3p^4 -> 8 divisors. This is ok with the statement.
if n is a product of two different prime numbers "p" and "q", the divisors are: ........ -> more than 8 divisors. we need 8 divisors.
n is a multiple of 3 positive integers
Answer is D. Each