The numbers {a, b,c} are three positive integers. If (a*b*c)/14

Question

The numbers {a, b,c} are three positive integers. If (a*b*c)/14 equals an integer and (b*c)/4 equals an integer, what is the smallest possible integer value of a?

A. 1
B. 2
C. 4
D. 7
E. 14

Papai21 · Answer

Given, (a*b*c)/14 = integer

Let, (a*b*c)/14 = k1

=> (a*b*c) = 2 * 7 * k1
 => a * 4k2 = 2 * 7 * k1 (As b* c = 4 * k2 given)
 => a = (7/2) * (k1/k2)
 => a = (7/2) * k (k is a positive integer)

As a is an integer so, minimum value of a is 7 when k = 2

So IMO ANSWER: D

Showmeyaa · Answer

\frac{a*b*c}{2*7*1}  = integer
 \frac{b*c}{4}  = integer
So, let b*c be = 28
So, a * b * c = 28
 a_{min}  = 1
IMO, (A)!

Suhasi · Answer

chetan2u  please help with this one's solution. IMO, it is D but I'm still unsure.
"b*c" must have at least two 2's.
14 has just one 2. So, "a" has to be 7.

chetan2u · Answer

Suhasi 
If you look at the expressions given, there is none restricting anything on a, for that matter on b and c individually too. Only restriction is that they are positive and integers. So 1 to infinity is the possibility for each.

Let me explain:-

\frac{(a*b*c)}{14}  is an integer: This does not mean that a*b *c is not divisible by 1000 or 2654321 and so on.
 It merely tells us that at least one of the three, a, b and c, is surely a multiple of 2 and a multiple of 7.

\frac{(b*c)}{4}  is an integer
 It merely tells us that at least one of b and c is surely even and could be a multiple of 4 too.

Combined
We can say that a*b*c is surely a multiple of LCM(14,4) or 28, and at least one of b or c >1.
So a could be as less as 1 or as great as 1000000000...

Even b or c individually can be 1, since we could take c or b respectively to be multiple of 28.

Vibhatu · Answer

Read the question carefully, that ask for the smallest value of a, not the smallest value of abc. If they ask fir the smallest value of abc, then a should be 7, otherwise for this question a is 1.
 Answer is A

Abolnasr · Answer

This is a weird question,
if the question was "What is the least possible value of a that makes this equation correct in 100% of cases?" then it MUST be 14 because in any other number less than 14, it's possible that b*c product is not divisible by any other number, so to eliminate division there must be 14 in the nominator to be canceled with the denominator.

IMO the question is stated very poorly.

Shaik98 · Answer

When the question uses the word " Possible ", even 1 case will suffice. We don't need it to be true all the time.

Abc = 14k ( K be a positive integer )
If A = 1 
Bc = 14k 
7 x 4 = 14 X 2 ( Assuming values for B,C,K just to suffice the condition)

BC/4 = (7X4)/4 = 7 ( Hold true as 7 is just another positive integer)

Hence 1 could " possibly " be the least possible positive integer. ( We don't have to prove for all cases as the question clearly states the word "Possible" ).

Posted from my mobile device

chamhi · Answer

Honestly the question is poorly worded. Smallest "possible" value is, in my view, clearly intended to be read as meaning the smallest, but always valid, value of a given the setup. IMO D.

aksjain2401 · Answer

The answer is 7 only according to mathematical logic. The basis for answer being 1 would have been present, had the question been what "could" be the smallest value of a. If GMAT wants an open ended interpretation of b*c like previous responses have assumed that if b*c = 28 , then a(minimum) = 1 ; then it specifically gives language hints by using words like could or maybe, which is not the case in this question, so mark 7.

The numbers {a, b,c} are three positive integers. If (abc)/14

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