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If a represents a single digit and the four digit

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If a represents a single digit and the four digit

by VJesus12 » Wed Jun 20, 2018 2:30 am

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If a represents a single digit and the four digit integer a12a is divisible by 6, what is a?

(A) 2
(B) 3
(C) 4
(D) 6
(E) 7

The OA is the option D.

Could anybody explain this PS question to me? Should I try option by option? What should I do?
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by Vincen » Wed Jun 20, 2018 3:29 am
Hello Vjesus12.

Let's take a look at your question.

We know that the number $$a12a$$ is divisible by 6.

Since 6=2*3, then this implies that the number is divisible by 2 and by 3.

Since the number is divisible by 2 then the number is even. This implies that a=2, 4, 6 or 8.

Now, since the number is divisible by 3, then the sum of its digits must be divisible by 3. That is to say, $$a+1+2+a=2a+3$$ is divisible by 3.

The only option that works is a=6 because we get that the sum is equal to 15.

Therefore, the correct option is [spoiler]D=6[/spoiler].

I hope it helps. <i class="em em-smiley"></i>
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by [email protected] » Wed Jun 20, 2018 9:23 pm
Hi VJesus12,

We're told that "A" represents a single digit and the four digit integer A12A is divisible by 6. We're asked for the value of A. This question can be solved with a mix of Number Property Rules and TESTing THE ANSWERS.

To start, if you weren't sure how to do anything else, then you could simply plug in each of the 5 answer choices and see which one ended up leading to a 4-digit number that is divisible by 6. Once you found that result, then you could stop working. There are some Number Property 'shortcuts' though, so we can use those to eliminate a couple of answers immediately.

For a number to be evenly divisible by 6, it must be divisible by BOTH 2 and 3. To be divisible by 2, the number must be EVEN. That would not occur with Answers B or E though, so we can eliminate both of those. For a number to be evenly divisible by 3, the DIGITS of that number must SUM to a number that is evenly divisible by 3. For example:
15 is divisible by 3 because 1+5 = 6. Since 6 is divisible by 3, we know that 15 is divisible by 3.
16 is NOT divisible by 3 because 1+6 = 7 is NOT divisible by 3.

Based on the remaining 3 answer choices, we have to consider these 3 values:
2122
4124
6126

Only one of those answers consists of digits that sum to a multiple of 3....

Final Answer: D

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by Jeff@TargetTestPrep » Thu Jun 21, 2018 4:13 pm
VJesus12 wrote:If a represents a single digit and the four digit integer a12a is divisible by 6, what is a?

(A) 2
(B) 3
(C) 4
(D) 6
(E) 7
We see that a12a is even, so it is divisible by 2. Because a12a is divisible by 6, we know that it must also be divisible by 3.

Recall that when a number is divisible by 3, the sum of its digits is a multiple of 3. Thus, the sum a + 1 + 2 + a must be a multiple of 3. So we see that a must be 6, since 6 + 1 + 2 + 6 = 15, which is a multiple of 3, and 6126 is even.

Answer: D

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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