BTGmoderatorDC wrote: ↑Sun Jan 31, 2021 4:39 pm
If p is a positive integer, is p^2 divisible by 96?
(1) p is a multiple of 8.
(2) p^2 is a multiple of 12.
OA
C
Source: Manhattan Prep
Target question: Is p² divisible by 96?
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 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 96 = (2)(2)(2)(2)(3), we can rephrase the target question as:
REPHRASED target question: Are there four 2's and one 3 hiding in the prime factorization of p² ?
Aside: the video below has tips on rephrasing the target question
Statement 1: p is a multiple of 8
In other words, p is divisible by 8
8 = (2)(2)(2)
So, we know that there are at least three 2's hiding in the prime factorization of p
This also tells us that
there are SIX 2's hiding in the prime factorization of p²
Unfortunately this information is not sufficient to answer the target question.
Consider these two possible cases:
Case a: p = 8, in which case p² = 64. In this case, the answer to the target question is
NO, p² is NOT divisible by 96
Case b: p = 24, in which case p² = 576. In this case, the answer to the target question is
YES, p² IS divisible by 96
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: p² is a multiple of 12
12 = (2)(2)(3)
So, we know that there are at least two 2's and one 3 hiding in the prime factorization of p²
Unfortunately this information is not sufficient to answer the target question.
Consider these two possible cases:
Case a: p = 12, in which case p² = 144 (which is divisible by 12). In this case, the answer to the target question is
NO, p² is NOT divisible by 96
Case b: p = 24, in which case p² = 576 (which is divisible by 12). In this case, the answer to the target question is
YES, p² IS divisible by 96
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that
there are SIX 2's hiding in the prime factorization of p²
Statement 2 tells us that there is at least ONE 3 hiding in the prime factorization of p²
So, when we combine the two statements, we can be certain that
there are at least four 2's and one 3 hiding in the prime factorization of p²
Since we can answer the
REPHRASED target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
