If p, q, and r are positive integers such that p is a factor of q, and

Question

If p, q, and r are positive integers such that p is a factor of q, and p is a multiple of r, which of the following MUST be an integer ?

A. (p + q)/q

B. (q + p)/r

C. (q + r)/p

D. pr/q

E. q/(r + p)

RahulJain293 · Answer

R is basically the factor of both P & Q.

Hence, to get an integer, R has to be in the denominator so as to be able to divide completely without leaving any decimals.

Choice B tells us P & Q are added. And since both are multiples of R, the sum should also be a multiple of R. ie., P+Q is divisible by R.

(Ex., if we have 4 & 6 which are multiples of 2. The sum of 4&6, 10, will also be a multiple of 2)

Hence B.

Adarsh_24 · Answer

q=p*a
p=r*b
(q+p)/r=(p.a+p)/r=p(a+1)/r=b(a+1)

Ans B

egmat · Answer

First, translate the words into math!

"p is a factor of q" → q/p is an integer → q = p × (some integer) 
 "p is a multiple of r" → p/r is an integer → p = r × (some integer)

So we have a chain:  r divides p, and p divides q

This means:
• q/p = integer ✓
• p/r = integer ✓
• q/r = integer ✓ (since r divides p, and p divides q)

But watch out!  These are  NOT  necessarily integers:
• p/q (since p is smaller than or equal to q)
• r/p (since r is smaller than or equal to p)

Now let's check each option:

(A) (p + q)/q = p/q + 1 
p/q is NOT necessarily an integer (p ≤ q)
Example: p =  2 , q =  6  → (2+6)/6 = 8/6 =  not an integer  ✗

(B) (q + p)/r = q/r + p/r 
 Both q/r and p/r are integers! 
• p/r is an integer (given: p is a multiple of r)
• q/r is also an integer (since q = p × k, and p = r × m, so q = r × m × k)
Integer + Integer =  Always an integer  ✓

(C) (q + r)/p = q/p + r/p 
q/p is an integer, BUT r/p is  NOT  necessarily an integer
Example: r =  2 , p =  4  → r/p = 1/2 ✗

(D) pr/q 
Example: p =  2 , q =  6 , r =  1  → pr/q = 2/6 =  not an integer  ✗

(E) q/(r + p) 
Example: p =  2 , q =  4 , r =  1  → 4/(1+2) = 4/3 =  not an integer  ✗

Answer: B

chloreton · Answer

Assume good set of numbers. q=24, p=6 and r=3
p is a factor of q (Yes, 6 is a factor of 24)
p is a multiple of r (Yes, 6 is a multiple of 3)

Now we can substitute q, p & r in the option and only  B  satisfies.

If p, q, and r are positive integers such that p is a factor of q, and

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If p, q, and r are positive integers such that p is a factor of q, and

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