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When A and B are positive integers, is AB a multiple of 4?

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When A and B are positive integers, is AB a multiple of 4?  [#permalink]

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New post 12 Oct 2018, 07:34
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[Math Revolution GMAT math practice question]

When A and B are positive integers, is AB a multiple of 4?

1) The greatest common divisor of A and B is 6
2) The least common multiple of A and B is 30

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When A and B are positive integers, is AB a multiple of 4?  [#permalink]

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New post Updated on: 12 Oct 2018, 12:53
MathRevolution wrote:
[Math Revolution GMAT math practice question]

When A and B are positive integers, is AB a multiple of 4?

1) The greatest common divisor of A and B is 6
2) The least common multiple of A and B is 30


A*B = 4 * integer?

Statement 1) tells us GCD (A,B) = 6

If we let a = 6 and b = 6 we get a*b = 36 which is a multiple of 4.

Sufficient.

Statement 2) LCM of a and b = 30

30 = 2*3*5

Meaning a could be 5 and b could be 6 or any other way of getting 30.

2*15, 3*10, 5 * 6

LCM (5,6) = 30 it is not a multiple of 4
LCM (6,30) = 30, 6 * 30 = 180/4 = 45 a multiple of 4

Insufficient.

Answer choice A

Posted from my mobile device

Originally posted by Salsanousi on 12 Oct 2018, 09:08.
Last edited by Salsanousi on 12 Oct 2018, 12:53, edited 1 time in total.
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Re: When A and B are positive integers, is AB a multiple of 4?  [#permalink]

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New post 12 Oct 2018, 12:46
MathRevolution wrote:
[Math Revolution GMAT math practice question]

When A and B are positive integers, is AB a multiple of 4?

1) The greatest common divisor of A and B is 6
2) The least common multiple of A and B is 30

Another beautiful problem, Max. Congrats! (and kudos!)

\(A,B\,\,\, \ge 1\,\,\,{\rm{ints}}\)

\({{A \cdot B} \over 4}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}}\)

\(\left( 1 \right)\,\,\,GCD\left( {A,B} \right) = 6\,\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
\,A = 6M\,,\,\,\,M \ge 1\,\,{\mathop{\rm int}} \hfill \cr
\,B = 6N\,,\,\,\,N \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,{\rm{with}}\,\,\,\,M,N\,\,\,{\rm{relatively}}\,\,\,{\rm{prime}}\)

\(?\,\,\,\,:\,\,\,\,{{A \cdot B} \over 4}\,\,\, = \,\,\,{{6M \cdot 6N} \over 4}\,\,\, = \,\,\,9MN\,\,\, = \,\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)


\(\left( 2 \right)\,\,\,LCM\left( {A,B} \right) = 30\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {A,B} \right) = \left( {1,30} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {A,B} \right) = \left( {2,30} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.\)


The correct answer is (A), indeed.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: When A and B are positive integers, is AB a multiple of 4?  [#permalink]

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New post 13 Oct 2018, 07:11
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

When A and B are positive integers, is AB a multiple of 4?

1) The greatest common divisor of A and B is 6
2) The least common multiple of A and B is 30


Target question: Is AB a multiple of 4?

Useful property: If N is divisible by d, we can say that N = dk for some integer k
For example, if N is divisible by 5, we can say that N = 5k for some integer k


Statement 1: The greatest common divisor of A and B is 6
This means that A is divisible by 6, and B is divisible by 6.
Applying the above property, we can write A = 6k for some integer k, and B = 6j for some integer j
So, AB = (6k)(6j) = 36kj = (4)(9)(kj)
Aha! We can see that AB is a multiple of 4
So, the answer to the target question is YES, AB IS a multiple of 4
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of A and B is 30
There are several values of A and y that satisfy statement B. Here are two:
Case a: A = 10 AND B = 30 (the LCM of 10 and 30 is 30). In this case, AB = (10)(30) = 300. So, the answer to the target question is YES, AB IS a multiple of 4
Case b: A = 1 AND B = 30 (the LCM of 1 and 30 is 30). In this case, AB = (1)(30) = 30. So, the answer to the target question is NO, AB is NOT a multiple of 4
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Re: When A and B are positive integers, is AB a multiple of 4?  [#permalink]

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New post 14 Oct 2018, 17:53
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Asking if \(AB\) is a multiple of \(4\) is equivalent to asking if \(AB = 4k\) for some integer k.

Condition 1)
Since \(A = 6a = 2*3\)*a and \(B = 6b = 2*3*b\) for some integers \(a\) and \(b\), \(AB = 2^2*3^2*ab = 4*3^2ab.\)
Thus, \(AB\) is a multiple of \(4\) and condition 1) is sufficient.

Condition 2)
If \(A = 6\) and \(B = 5\), then \(lcm(A,B) = 30\), but \(AB = 30\) is not a multiple of \(4\), and the answer is ‘no’.
If \(A = 6\) and \(B = 10\), then \(lcm(A,B) = 30\), and \(AB = 60\) is a multiple of \(4\). The answer is ‘yes’.
Since we don’t have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A
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Re: When A and B are positive integers, is AB a multiple of 4?   [#permalink] 14 Oct 2018, 17:53
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