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coolvishu11: Junior | Next Rank: 30 Posts; Posts: 17; Joined: Tue Sep 04, 2007 7:45 am; Thanked: 1 times

by coolvishu11 » Sun May 10, 2009 4:24 pm

4. If the square root of p2 is an integer, which of the following must
be true?
I. p2 has an odd number of factors
II. p2 can be expressed as the product of an even number of prime
factors
III. p has an even number of factors

a· I
b· II
c· III
d· I and II
·e II and III

OA is D

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Source: Beat The GMAT — Problem Solving | Sun May 10, 2009 4:24 pm

DeepakR: Master | Next Rank: 500 Posts; Posts: 158; Joined: Tue Sep 30, 2008 8:47 amFollowed by: 1 member; Thanked: 23 times; GMAT Score:660

by DeepakR » Sun May 10, 2009 5:04 pm

Let p^2 = 16.

1. Factors of 16 = 1, 2, 4, 8, 16 = 5 numbers which are factors and hence #1 satisfies.

2. Prime factors of 16 = 2*2*2*2 which is the product of even number of prime factors of 16.

3. Does not satisfy due to #1

You can also check with number like 25

1. Factors of 25 = 1, 5, 25 = satisfies
2. Prime factors of 25 = 5 * 5 = we have 2 fives which are prime and even numbers (i.e two 5's)
3. Does not satisfy due to #1

Hence D.)

-Deepak

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sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Sun May 10, 2009 7:54 pm

Since p^2 is a perfect square and every perfect square will always have odd number of factors, I is always true.

Also, since p^2 contains odd of factors, when p^2 is written in the form of prime factorization, each and every prime factor will be raised to an even power and hence p^2 can always be written as even number of prime numbers. Hence, II is true.

I contradicts III

Thus I and II are true.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Mon May 11, 2009 3:56 am

sureshbala wrote: I contradicts III

I think you may have misread III; III says that p has an even number of factors, not that p^2 has an even number of factors, so I and III are not contradictory.

The only numbers which have an odd number of positive divisors are perfect squares. Since p^2 is a perfect square, I must be true. III may be true, and it may not; if p is also a perfect square (e.g. if p = 4, and p^2 = 16, or p = 9 and p^2 = 81), then p will have an odd number of divisors. If p is not a perfect square, then p will have an even number of divisors.

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