Is x a multiple of 72?

I think the answer is C.

Explanation:

Statement 1: Not every multiple of 16 is a multiple of 72. Ex: 80 is a multiple of 16 however not a multiple of 72. However, 144 is also a mutiple of 16 and is a multiple of 72. Not sufficient.

Statement 2: Similar to above reason. For ex: 90 is divisible by 18 but not a multiple of 72. Not Sufficient

Statement 1&2: The only values of x satisfying both statements are 144, 288 etc. which are a multiple of 16 and also divisible by 18. Hence, these values are also multiples of 72.



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-----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 a multiple of k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is a multiple of 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is a multiple of 5 because 70 = (2)(5)(7)
And 112 is a multiple of 8 because 112 = (2)(2)(2)(2)(7)
And 630 is a multiple of 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------

Target question: Is x a multiple of 72?
This is a good candidate for rephrasing the target question.

72 = (2)(2)(2)(3)(3)
REPHRASED target question: Are three 2's and two 3's hiding in the prime factorization of x?

Aside: the video below has tips on rephrasing the target question

Statement 1: x is a multiple of 16
16 = (2)(2)(2)(2)
So, this tells us that there are four 2's hiding in the prime factorization of x.
However, we can't be sure whether there are also two 3's hiding in the prime factorization of x.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x is divisible by 18
18 = (2)(3)(3)
So, this tells us that there is one 2 and two 3's hiding in the prime factorization of x.
This means we have enough 3's, however we can't be sure whether there are also three 2's hiding in the prime factorization of x
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT


Statements 1 and 2 combined
Statement 1 tells us that there are three 2's hiding in the prime factorization of x
Statement 2 tells us that there are two 3's hiding in the prime factorization of x
So when we COMBINE statements, the answer to the REPHRASED target question is YES, there are three 2's and two 3's hiding in the prime factorization of x
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

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