Target question: Is 3 a factor of the positive integer m?M7MBA wrote:Is 3 a factor of the positive integer m?
(1) 12 is a factor of 25m
(2) 12 is a factor of 15m
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 factor by k, then k is "hiding" within the prime factorization of N
Consider these examples:
3 is a factor of 24, because 24 = (2)(2)(2)(3), and we can clearly see the 3 hiding in the prime factorization.
Likewise, 5 is a factor of 70 because 70 = (2)(5)(7)
And 8 is a factor of 112 because 112 = (2)(2)(2)(2)(7)
And 15 is a factor of 630 because 630 = (2)(3)(3)(5)(7)
-----BACK TO THE QUESTION!---------------------
The above concept allows us to REPHRASE the target question as...
REPHRASED target question: Is there a 3 hiding in the prime factorization of m?
Statement 1: 12 is a factor of 25m
12 = (2)(2)(3)
In other words, statement 1 is telling us that there are two 2's and one 3 hiding in the prime factorization of 25m
25m = (5)(5)(m)
Since there are no 3's hiding in (5)(5), it must be the case that there's a 3 hiding in the prime factorization of m
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: 12 is a factor of 15m
12 = (2)(2)(3)
In other words, statement 2 is telling us that there are two 2's and one 3 hiding in the prime factorization of 15m
15m = (3)(5)(m)
As we can see, we already have a 3 hiding in the prime factorization of 15m, so we can't say for sure whether there's a 3 hiding in the prime factorization of m.
To more certain, consider these two counter-examples:
Case a: m = 4. This means 15m = (15)(4) = 60, and 12 IS a factor of 60. In this case, 3 is NOT a factor of m
Case b: m = 12. This means 15m = (15)(12) = 180, and 12 IS a factor of 180. In this case, 3 IS a factor of m
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
