Problem Solving

NandishSS wrote:If n is multiple of 5, and n = p²q, where p and q are prime numbers, which of the following MUST be a multiple of 25?

A) p²
B) q²
C) pq
D) p²q²
E) p³q

Let's test a few values that satisfy the given conditions (n is multiple of 5, and n = p²q, where p and q are prime numbers)

How about: p = 2 and q = 5.
In this case, n = (2²)(5) = 20, and 20 is a multiple of 5, which satisfies the given condition.

Now plug p = 2 and q = 5 into the answer choices...
A) 2² = 4. This is NOT a multiple of 25. ELIMINATE
B) 5² = 25. This IS a multiple of 25. KEEP
C) (2)(5) = 10. This is NOT a multiple of 25. ELIMINATE
D) (2²)(5²) = 100. This IS a multiple of 25. KEEP
E) (2³)(5) = 40. This is NOT a multiple of 25. ELIMINATE

So, the correct answer is either B or D.

Let's try a new set of values.
How about: p = 5 and q = 2.
In this case, n = (5²)(2) = 20, and 20 is a multiple of 5, which satisfies the given condition.

Now plug p = 5 and q = 2 into the REMAINING answer choices...
B) 2² = 4. This is NOT a multiple of 25. ELIMINATE

NOTE: At this point, we can safely conclude that the correct answer is D.
But let's try D for "fun"...
D) (5²)(2²) = 100. This IS a multiple of 25.

Answer: D

If n is multiple of 5, and n = p²q, where p and q are prime numbers, which of the following MUST be a multiple of 25?

A) p²
B) q²
C) pq
D) p²q²
E) p³q

Another approach:

If p and q are prime numbers, and p²q is divisible by 5, then either p = 5, q = 5 or they both equal 5.

We're looking for an expression that MUST be divisible by 25, which means there must be TWO 5's "hiding" in the expression.

A) p²
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 2 and q = 5, in which case p² is NOT divisible by 25
ELIMINATE A

B) q²
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 5 and q = 2, in which case q² is NOT divisible by 25
ELIMINATE B

C) pq
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 5 and q = 2, in which case pq is NOT divisible by 25
ELIMINATE C

D) p²q²
YES, we can be certain that there are TWO 5's "hiding" in this expression.
If p = 5, then p²q² = 25q², which is DEFINITELY divisible by 25
If q = 5, then p²q² = 25p², which is DEFINITELY divisible by 25

E) p³q
We cannot be certain that there are TWO 5's "hiding" in this expression.
It could be the case that p = 2 and q = 5, in which case p³q is NOT divisible by 25
ELIMINATE E

Answer = D

Cheers,
Brent

Hi NandishSS,

This question is built around a couple of Number Properties and can be solved by TESTing VALUES.

To start, we're told two things about N...
1) N is a multiple of 5
2) N = (P)(P)(Q)

Since N is a multiple of 5, at least one of it's prime factors MUST be a 5. We're told that P and Q are both PRIME, which means that P or Q or both will be a multiple of 5. This is an interesting point, since the question asks which of the following MUST be a multiple of 25 (meaning - which of these answers will ALWAYS be a multiple of 25 no matter how many different examples you can come up with?). As such, we will have to consider a couple of different possibilities...

IF...
P = 5
Q = 2
N = 50
We can eliminate answers B and C.

IF....
P = 2
Q = 5
N = 20
We can eliminate answers A and E.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

NandishSS wrote:If n is a multiple of 5, and n=(p^2)q where p and q are prime, which of the following must be a multiple of 25?

A.p^2
B.q^2
C.pq
D.(p^2)(q^2)
E.(p^3)qm

A common phrase that is used on the GMAT is the word must. In this question, we are asked which of the following must be a multiple of 25. This means that one of our answer choices will always be a multiple of 25, no matter what. It is our job to determine which one, based on the given information.

We are given that n is a multiple of 5, n = (p^2)q, and that p and q are prime numbers.

Because n is a multiple of 5, a prime number, we know that either p or q is 5. Let's now analyze each answer choice to determine which one MUST (in all cases) be a multiple of 25.

A) p^2

If p = 3, then p^2 = 9 is not a multiple of 25. Answer choice A is not correct.

B) q^2

If q = 3, then q^2 = 9 is not a multiple of 25. Answer choice B is not correct.


C) pq

If p = 5 and q = 3 (or vice versa), pq = 15 is not a multiple of 25. Answer choice C is not correct.

D) (p^2)(q^2)

Regardless of which values we select for p and q, since we know that either p or q is 5, (p^2)(q^2) will always be a multiple of 25. If this is too difficult to see, let's use numbers.

If p = 5 and q = 3, (p^2)(q^2) = (25)(9) is a multiple of 25.

If p = 3 and q = 5, (p^2)(q^2) = (9)(25) is also a multiple of 25.

Answer choice D is correct.

For practice, let's analyze answer choice E.

E) (p^3)q

If p = 3 and q = 5, then (p^3)q = 135 is not a multiple of 25. Answer choice E is not correct.

Answer: D