Divisibility yes/no question

[infiniti007]

Is integer x divisible by 24?

1.) x is divisible by 6.
2.) x is divisible by 4.

I understand how to do this which "trying numbers". I would like to understand how to approach this using prime factors.

For instance: 24 is equal to: 3*2*2*2.
Statement (1) provides that x is divisible by 3*2
Statement (2) provides that x is divisible by 2*2

What is the gap in logic to think that by combining them, x would have the prime factors it needs to be divisible by 24?

[DavidG@VeritasPrep]

Imagine that we have a box marked 'x' and we want to know if that box, which we know is full of prime numbers, contains at least three 2's and one 3.

One person peeks into the box and sees one 2 and one 3. A second person peeks into the box and sees two 2's.

After conferring with our two witnesses we can't deduce that there are three 2's contained in the box, because it's possible that our hypothetical people saw the same 2. All we know for sure is that the box contains two 2's and one 3.

(This is why the least common multiple of 6 and 4 is 2*2*3 = 12.)

[Brent@GMATPrepNow]

Nicely put, David!

[Brent@GMATPrepNow]

Is integer x divisible by 24?

1.) x is divisible by 6.
2.) x is divisible by 4.

Target question: Is integer x divisible by 24?

-------------------------------
ASIDE: A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
----------------------------

Since 24 - (2)(2)(2)(3), we can REPHRASE the target question....
REPHRASED target question: Are there three 2's and one 3 hiding in the prime factorization of x?

Aside: for more on this concept, see our free video: https://www.gmatprepnow.com/module/gmat- ... /video/825

Statement 1: x is divisible by 6.
6 = (2)(3), so we can conclude that x has at least ONE 2 and ONE 3 hiding in its prime factorization.
Of course, there MIGHT be three 2's and one 3 hiding in the prime factorization of x. But we can't say for sure.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT


Statement 2: x is divisible by 4.
4 = (2)(2), so we can conclude that x has at least TWO 2's hiding in its prime factorization.
Of course, there MIGHT be three 2's and one 3 hiding in the prime factorization of x. But we can't say for sure.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that that x has at least ONE 2 and ONE 3 hiding in its prime factorization.
Statement 2 tells us that that x has at least TWO 2's hiding in its prime factorization.
Combined, we can be certain that x has at least TWO 2 and ONE 3 hiding in its prime factorization.
Of course, there MIGHT be three 2's and one 3 hiding in the prime factorization of x. But we can't say for sure.

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

[sandipgumtya]

David:Nice approach.Out of the box thinking...

[Max@Math Revolution]

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


Is integer x divisible by 24?

1.) x is divisible by 6.
2.) x is divisible by 4.


==> In the original condition we have 1 variable (x), and we need to match the number of variable and equation. Since there is 1 equation each in 1) and 2), the answer is likely D.

In case of 1), the answer is x=24 yes, x=12 no thus it is NOT sufficient. In case of 2), the answer is x=24 yes, x=12 no thus it is not sufficient. Using both 1) & 2), the answer is still x=24 yes, x=12 no, therefore it is not sufficient. Therefore the answer is E.

Generally if the original condition has 1 variable, there's 60% chance that the answer is D.



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