gana wrote:If n is a multiple of 5 and n=p^2q where p and q are prime numbers, which of the following must be a multiple of 25. ?
A. p^2
B. q^2
C. pq
D. p^2q^2
E. p^3q
Solve must be true questions by plugging in numbers and eliminating answer choices which are not MUST. Start by defining what you're looking for: the right answer choice MUST be a multiple of 25. the flip side of that is that there are four answer choices which CAN be a multiple of 25 (for some prime values of p and q), but don't have to be. The strategy is to find the four wrong answer choices by plugging in values for p and q, and eliminating those which are not multiples of 25 - thus proving that they're not MUST.
n has to be a multiple of 5, so at least one of p and q must be a multiple of 5 - if p and q are NOT multiples of 5 (for example, 2 and 3) we get n=2^2*3 = 12 which is not a multiple of 5.
Let's go with p=5, p=2. Plug these values into the answer choices:
A 2^2=4 is not a multiple of 25. Eliminate.
B 5^2 IS a multiple of 25. Doesn't mean it's the right answer though - just that we can't eliminate it YET.
C pq=2*55=10. Not a multiple of 25. Eliminate
D p^2q^2 = 5^2*2^2 = 25*4 = 100 IS a multiple of 5. Cannot eliminate yet.
E p^3q = 5^3*2 = 125*2 = 250 IS a multiple of 25. Can't eliminate yet.
So we're still left with B, D and E. plug in a second set into these answer choices and try to shake off those that will not give you a multiple of 25. For example, if we reverse the Plug ins p=2, q=5, B and E will no longer give you a multiple of 25, but D still will, so D is the right answer by POE.
The benefit of this approach is that it doesn't require any higher math concepts - just plug in and eliminate until only one answer choice remains, which has to be the right answer choice, and you needn't worry about why. It does require investing some thought into what's the grounds for elimination: if we want a "must be 25", we're looking for
counter examples: values of p and q that will NOT give a multiple of 25 for the four wrong answer choices.
In retrospect, D seems obvious: if p or q or both have to be multiples of 5, then p^2q^2 will definitely be a multiple of 5^2=25. But this is not at all obvious when looking at five answer choices together: plugging in and eliminating is a way of letting D stick out from the crowd.
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