If positive integer x is a multiple of 14 and 12, which of the followi

Question

If positive integer x is a multiple of 14 and 12, which of the following statements must be true?

I. x is divisible by 8
II. x is divisible by 28
III. The greatest common divisor of x and 9 is 3

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

sach24x7 · Accepted Answer

LCM Of 14 and 12 is 84.

So x would be multiple of 84 such as 84*1, 84*2, ....,84*8 etc

1) x may or may not be divisible by 8
2) x is surely divisible by 28 because 84 is divisible by 28.
3) GCF of x and 9 can be 3 or or 9 based on what value of x in above mentioned possibilities.

So 2 only must be true.

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nick1816 · Answer

X is a multiple of 12 and 14, Then X must be in the form of 2^2*3*7*k, where k is an positive integer.

1. If k is odd, X can't be multiple of 8
2. X=2^2*3*7*k= 28*3k, hence always a multiple of 28
3. If k is multiple of 3, then GCD(x,9)= 9

Only 2nd statement must be true

ScottTargetTestPrep · Answer

We recall that if a number is a multiple of two numbers, it’s also a multiple of the LCM of the two numbers. The LCM of 14 and 12 is 7 * 4 * 3 = 84. So x is a multiple of 84.

We also recall that if a is a multiple of b, then it’s also a multiple of any factors of b.

Thus, we see that x MUST BE divisible by 28 (since 28 is a factor of 84) but not necessarily by 8 (since 8 is not a factor 84). Thus, we see that statement I is not necessarily true, and statement II must be true.

The GCF of x and 9 does not have to be 3. For example, if x = 84 * 3 = 252, then the GCF of x and 9 is 9. Thus, statement III is not necessarily true.

Answer: B

HouseStark · Answer

The question would be more logical if the question were

If positive integer x is divisible by both 14 and 12, which of the following statements must be true?

I started thinking 12*14

whose prime factorization is 7*3* 2^3

Do we get this kinda question in GMAT??

Adit_ · Answer

Its easy to bring it down to options B and E only.
Option E can easily checked by taking factors of 84 like 3*28 = 84 why I did this is because if we multiple another 3 it is automatically a multiple of 9 and since we have 3 as its factors for sure, highest factor of x and 9 need NOT be 3. Hence NOT always true.

Answer:  Option B

egmat · Answer

Let's break this down step by step.

Since x is a multiple of both  14  and  12 , x must be a multiple of their  LCM (Least Common Multiple) .

14  =  2  ×  7 
 12  =  2 2 ×  3

LCM ( 14 ,  12 ) =  2 2 ×  3  ×  7  =  84

So x must be a multiple of  84 . This means x could be  84 ,  168 ,  252 ,  336 , and so on.

Now let's test each statement:

Statement I:  x is divisible by  8 
 8  =  2 3. But  84  only contains  2 2 (two factors of  2 , not three). So x =  84  is NOT divisible by  8 .  Statement I does NOT have to be true.

Statement II:  x is divisible by  28 
 28  =  2 2 ×  7 . Since  84  =  2 2 ×  3  ×  7 , every multiple of  84  automatically contains  2 2 and  7 . In fact,  84  ÷  28  =  3 , a whole number. So every multiple of  84  is also a multiple of  28 .  Statement II MUST be true.

Statement III:  The  GCD  of x and  9  is  3 
 9  =  3 2. When x =  84  =  2 2 ×  3  ×  7 , the  GCD ( 84 ,  9 ) =  3 . Looks good so far. But when x =  252  =  84  ×  3  =  2 2 ×  3 2 ×  7 , the  GCD ( 252 ,  9 ) =  9 , NOT  3 .  So Statement III does NOT have to be true — it fails when x has more than one factor of 3.

Only Statement II must always be true.

Answer: B

Key principle:  When a problem says 'x is a multiple of A and B,' always find the  LCM  first — that tells you the minimum guaranteed prime factorization of x. Then test each statement against that factorization. For 'must be true' questions, also check larger multiples to see if a statement can break.

If positive integer x is a multiple of 14 and 12, which of the followi

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If positive integer x is a multiple of 14 and 12, which of the followi

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