Brent@GMATPrepNow wrote:Is positive integer p even?
(1) 4p has twice as many positive divisors as p has
(2) 8p has 3 positive divisors more than p has
Answer:
A
Source:
www.gmatprepnow.com
Difficulty level: 700+
------ASIDE---------------------
Here's a useful rule:
If the
prime factorization of N = (p^
a)(q^
b)(r^
c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (
a+1)(
b+1)(
c+1)(etc) positive divisors.
Example: 14000 = (2^
4)(5^
3)(7^
1)
So, the number of positive divisors of 14000 = (
4+1)(
3+1)(
1+1) =(5)(4)(2) = 40
-----------------------------------
Target question: Is positive integer p even?
Statement 1: 4p has twice as many positive divisors as p has
Since p is a positive INTEGER, we know that p is either EVEN or ODD
I'll show that
p cannot be odd, which will allow us to conclude that p must be even.
If p is ODD, then the prime factorization of p will consist of ODD primes only.
We can write: p = (some odd prime^
a)(some odd prime^
b)(some odd prime^
c)....
So, the number of positive divisors of p = (
a+1)(
b+1)(
c+1)...
Let's let k = (
a+1)(
b+1)(
c+1)...
That is, k = the number of positive divisors of p
Now let's examine the prime factorization of
4p
4p = (
2^2)(
a+1)(
b+1)(
c+1)...
So, the number of positive divisors of
4p = (
2+1)(
a+1)(
b+1)(
c+1)...
= (
3)(
a+1)(
b+1)(
c+1)...
= (
3)(k)
So, p has k divisors, and 4p has 3k divisors.
In other words, 4p has THREE TIMES as many divisors as p.
HOWEVER, we need 4p to have TWICE as many divisors as p.
So, we can conclude that p CANNOT be odd, which means
p must be even
Aside: For example, if p = 2, then it has 2 divisors, and 4p = 8, which has 4 divisors. So, 4p has TWICE as many divisors as p
Statement 2: 8p has 3 positive divisors more than p has
There are several values of p that satisfy statement 2. Here are two:
Case a: p = 1, which means 8p = 8. 1 has 1 divisor (1), whereas 8 has 4 divisors (1, 2, 4, 8). So, 8p has 3 positive divisors more than p has. In this case, the answer to the target question is
NO, p is NOT even
Case b: p = 2, which means 8p = 16. 2 has 2 divisors (1, 2), whereas 16 has 5 divisors (1, 2, 4, 8, 16). So, 8p has 3 positive divisors more than p has. In this case, the answer to the target question is
YES, p is even
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
