Bunuel wrote:
Is the positive integer n a multiple of 24 ?
(1) n is a multiple of 4.
(2) n is a multiple of 6.
Target question: Is the positive integer n a multiple of 24 ?This is a good candidate for
rephrasing the target question.
-----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 NConsider 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!---------------------
Since 24 = (2)(2)(2)(3), we can rephrase the target question as follows...
REPHRASED target question: Are three 2's and one 3 hiding in the prime factorization of n?Aside: the video below has tips on rephrasing the target question Statement 1: n is a multiple of 4. 4 = (2)(2), so statement 1 is telling us that there are two 2's hiding in the prime factorization of n.
Of course, there COULD be additional 2's (and 3's for that matter) hiding in the prime factorization of n.
Given this, there's no way we can answer the target question with certainty.
If you're not convinced, consider these two possible cases:
Case a: n = 12 (which is a multiple of 4). In this case, the answer to the target question is
NO, n is not divisible by 24Case b: n = 48 (which is a multiple of 4). In this case, the answer to the target question is
YES, n is divisible by 24Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n is a multiple of 66 = (2)(3), so statement 2 is telling us that there is one 2 and one 3 hiding in the prime factorization of n.
Of course, there COULD be additional 2's and 3's for that matter hiding in the prime factorization of n.
Given this, there's no way we can answer the target question with certainty.
Consider these two possible cases:
Case a: n = 12 (which is a multiple of 6). In this case, the answer to the target question is
NO, n is not divisible by 24Case b: n = 48 (which is a multiple of 6). In this case, the answer to the target question is
YES, n is divisible by 24Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined IMPORTANT: Notice that I was able to use the
same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: n = 12 (which is a multiple of 4 and 6). In this case, the answer to the target question is
NO, n is not divisible by 24Case b: n = 48 (which is a multiple of 4 and 6). In this case, the answer to the target question is
YES, n is divisible by 24Since we cannot answer the
target question with certainty, the combined statements are NOT SUFFICIENT
Answer: E
Cheers,
Brent
RELATED VIDEO