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# If n and p are different positive prime numbers, which of

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Manager
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If n and p are different positive prime numbers, which of [#permalink]  26 Feb 2006, 09:39
If n and p are different positive prime numbers, which of the integers ,
and np has (have) exactly 4 positive divisors?
(A) n4 only
(B) p3 only
(C) np only
(D) n4 and np
(E) p3 and np
Intern
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Re: Problem Solving Help [#permalink]  26 Feb 2006, 10:42
Avis wrote:
If n and p are different positive prime numbers, which of the integers ,
and np has (have) exactly 4 positive divisors?
(A) n4 only
(B) p3 only
(C) np only
(D) n4 and np
(E) p3 and np

E?

Here's why:
lets take two prime numbers: n=2 and p=11.
np=22 has 1, 2, 11, and 22 as the only divisors.
n4=8 has 1, 2, 4 and 8 as the only divisors
p3=33 has 1, 3, 11, 33 as the only divisors

So looks like all have exactly 4. But lets take another prime number combination. n=3 and p=5
np=15 has 1, 3, 5 and 15 as the only divisors
n4=12 has 1, 2, 3, 4, 6 and 12 as the divisors (Notice > 4 divisors)
p3=15 has 1, 3, 5 and 15 as the only divisors

So n4 cannot always have exactly 4 divisors.
Manager
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I pick C.

If n= 5 and p=3, then n4= 20 which would mean 1, 20, 4, 5, 10, 2 and p3 = 9 which is 1,3 and 9.

So np = 15 still has 1,3, 5 , 15 as its divisors. Np only has 4 positive divisors
GMAT Club Legend
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A)n4 will have divisors --> 1,2,n,n4 --> Could be 4, but if n = 2, then will be 3
B)same as A
C)np --> 1,n,p,np
D)n4 and np --> 1,n,p,n4,np --> could be as much as 5
E)p3 and np --> 1,3,n,p,p3,np --> could be as many as 6

I go with C
Manager
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i try to use numbers , long approach

Anyone has a shortcut ?

Thanks
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when there is a will there is a way

best regards

Manager
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Guys, I really did not understand the question. Does np indicate n*p and n4 indicate n*4?
If np = n*p, it is straight C. Otherwise, no choice is correct.
Senior Manager
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Nice and tricky question Avis.

Four factors of NP are 1, n, p, Np.

Manager
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ONE question.

If n4 means n*4, I think n4 also has 4 factors, which is 1, n, 2n, and 4n.

Am I misdirected?
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GMAT by 8th JUL

VP
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foolbox wrote:
ONE question.

If n4 means n*4, I think n4 also has 4 factors, which is 1, n, 2n, and 4n.

Am I misdirected?

You should also consider 2 and 4, so there are six factors, but the Q ask specifically for four.
Director
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Yup. C is the answer... explanation as discussed above.
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