NandishSS wrote:If 4 is a factor of positive integer x and 9 is a factor of positive integer y, is 42 a factor of xy?
(1) 14 is a factor of x.
(2) 25 is a factor of y.
Target question: Is 42 a factor of xy?
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)
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Since 42 = (2)(3)(7), the target question can be REPHRASED as...
REPHRASED target question: Is there a 2, and 3 and a 7 hiding in the prime factorization of xy?
Given: 4 is a factor of positive integer x and 9 is a factor of positive integer y
If 4 is a factor of x, then x = 4k for some integer k
If 9 is a factor of y, then y = 9j for some integer j
So, xy = (4k)(9j) = (2)(2)(3)(3)jk (for some integer jk)
Notice that
xy already (before we even examine each statement) has a 2 and a 3 hiding in its prime factorization.
The ONLY missing piece is a 7.
So, IF we get some information that tells us that there's a 7 hiding in the prime factorization of xy, then we can be certain that xy is divisible by 42.
This means we can REPHRASE our target question even more....
RE-REPHRASED target question: Is there a 7 hiding in the prime factorization of xy?
Statement 1: 14 is a factor of x
Since 14 = (2)(7), we know that there's a 2 and a 7 hiding in the prime factorization of x.
This also means there's a 2 and a 7 hiding in the prime factorization of xy.
More importantly,
there's a 7 hiding in the prime factorization of xy.
Since we can answer the
RE-REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: 25 is a factor of y
Since 25 = (5)(5), we know that there are two 5's hiding in the prime factorization of y.
IMPORTANT: This does not mean there are no 7's hiding in the prime factorization of y.
We just can't be sure.
As such,
we cannot definitively say whether or not there are any 7's hiding in the prime factorization of xy
Since we cannot answer the
RE-REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT
Answer =
A
Cheers,
Brent
