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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

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

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

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.