Taniuca wrote:If P,Q, and R are positive intergers such that q is a factor of r, and r is a multiple of p, which of the following must be and interger?
a) (p+q)/r
b) (r+p)/q
c) p/q
d) pq/r
e) r(p+q)/ pq
I chose P=3, Q=2 and R=6 that gives me the right answer d. What do I have wrong on my selection? the right choise is E. In advance, Thanks!
When a question asks
which of the answers must be something, be prepared to
plug in more than once. The goal as you plug in values is to disprove each answer choice: to
show that the answer choice doesn't have to be true.
Keep plugging in values until four answers have been eliminated.
In the problem above, we have to satisfy the conditions of the problem: r is a multiple of p, and q is a factor of r. In simpler terms, r is a multiple both of p and of q.
Let's start with your values: p=3, q=2, r=6.
A) (p+q)/r = (3+2)/6 = 5/6. Not an integer. Eliminate A.
B) (r+p)/q = (6+3)/2 = 9/2. Not an integer. Eliminate B.
C) p/q = 3/2. Not an integer. Eliminate C.
D) pq/r = 3*2/6 = 1. An integer. Hold onto D.
E) r(p+q)/pq = 6(3+2)/(3*2) = 30/6 = 5. An integer. Hold onto E.
Since two answers remain, we have to plug in different values.
Now let's think about how we might disprove D, which says that pq/r = integer. If we make p and q small and r larger, we'll get a fraction and prove that D doesn't have to be an integer. Let's try p=1, q=1, r=2. These values satisfy the conditions because r=2 is a multiple both of p=1 and of q=1.
D) pq/r = 1*1/2 = 1/2. Not an integer. Eliminate D.
The correct answer is E.
Since we eliminated A, B, C and D, we shouldn't waste valuable time checking E. But for the skeptical among us, if p=1, q=1 and r=2:
E) r(p+q)/pq = 2(1+1)/(1*1) = 4.
A little algebra proves why E has to be an integer:
E) r(p+q)/pq = (rp + rq)/pq = rp/pq + rq/pq = r/q + r/p = integer + integer, because r is a multiple both of q and of p. Since integer + integer = integer, E is the answer choice that must be an integer.
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