If a, b and c are positive integers such that a is a factor of b, and

Question

If a, b and c are positive integers such that a is a factor of b, and a is a multiple of c, which of the following is NOT necessarily an integer?

A. bc/a

B. (a + b)/c

C. (c - b)/a

D. (a + c)/c

E. 2bc/a

am0804 · Answer

IMO C.

Here's how I would break it down: 
"a is a factor of b" -> b = am (let m be a non zero integer) 
"a is a multiple of c" -> a = ck ( k is again a non zero integer) 
And combining both: b = ckm

Now we can simply plug in the values and see which ones will always be an integer.

Some are easy to detect for eg bc/a & 2bc/a because 'a' is a factor of 'b.'

C) (c-b)/a = (c-ckm)/ ck -> we will still have 'k' left in the denominator thus not necessarily an integer.

Posted from my mobile device

BrentGMATPrepNow · Answer

a is a factor of b  
This means  b/a  = j (where j is some integer)

a is a multiple of c  
This means  a/c  = k (where k is some integer)

This also means that (b/a)(a/c) = jk
Simplify to get:  b/c  = jk
If j and k are integers, we know that jk is an integer

A. bc/a 
We already know that:  b/a  = j (where j is some integer)
If we multiply both sides of the equation by c we get: c b/a  = jc 
Since j and c are both integers, we know that jc is an integer, which means bc/a is ALWAYS an integer

B. (a + b)/c 
(a + b)/c =  a/c  +  b/c 
= k + jk
= some integer (we already know k and jk are integers, it's also true that k + jk is an integer

C. (c - b)/a 
(c - b)/a =  c/a  -  b/a 
We know that  b/a  is an integer, but we don't know anything about  c/a 
So let's test some values
From the given information, it COULD be the case that a = 4, b = 12 and c = 2
This means (c - b)/a = (2 - 12)/4 = -10/4 = -2.5, and -2.5 is NOT an integer

Answer: C

Cheers,
Brent

Kinshook · Answer

Asked: If a, b and c are positive integers such that a is a factor of b, and a is a multiple of c, which of the following is NOT necessarily an integer?

b = k*a = k*m*c; a = m*c ; where k & m are positive integers

A. bc/a = (b/a)*c = k*c ; An integer

B. (a + b)/c= (a/c + b/c) = m + k*m = m(1+k) ; An integer

C. (c - b)/a = (c/a-b/a) = 1/m - k ; Not necessarily an integer

D. (a + c)/c = a/c + 1 = m + 1; An integer

E. 2bc/a = 2kc ; An integer

IMO C

cjcalero24 · Answer

I had an understanding of the theory behind this question but was having difficulty conceptualizing it so I used smart numbers that fit the criteria given.

a=8 b=16 c=4

Once I had these numbers it becomes clear that option c is not an integer

(4-16)/8 = -12/8

If a, b and c are positive integers such that a is a factor of b, and

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If a, b and c are positive integers such that a is a factor of b, and

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