leumas wrote:If k, m, and t are positive integers and (k/6)+ (m/4) = (t/12),
do t and 12 have a common factor greater than 1 ?
(1) k is a multiple of 3.
(2) m is a multiple of 4.
Here's another question we can solve using prime factorization (I just solved this divisibility question using prime factorization:
https://www.beatthegmat.com/divisibility-t91797.html)
First, for questions involving multiples, we can say:
If N is a multiple of w, then w is "hiding" within the prime factorization of N
Examples:
24 is a multiple of
3 <--> 24 = 2x2x2x
3
70 is a multiple of
5 <--> 70 = 2x
5x7
330 is a multiple of
6 <--> 330 =
2x3x5x11
Okay, let's begin by taking the target question "
Do t and 12 have a common factor greater than 1?" and rewrite it using prime factorization. We can reword it as "
Do the prime factorizations of t and 12 share a number greater that 1?"
Finally, before we examine the statements, let's take the given information, (k/6)+ (m/4) = (t/12), and make it "nicer" by multiplying both sides by 12 to get 2k + 3m = t
Statement 1: k is a multiple of 3.
In other words,
3 is hiding in the prime factorization of k
So, we know that k =
(3)(?)(?)(?)...
Aside: notice that we may or may not have primes other than the 3 in the prime factorization of k. All we can be certain of is that there is one 3 within the prime factorization
From here, we'll take our given information, 2k + 3m = t and replace k with
(3)(?)(?)(?) to get: (2)
(3)(?)(?)(?) + 3m = t
At this point, we can factor out a 3 to get:
3[(2)(?)(?)(?) + m] = t, which means 3 is a divisor of t
If 3 is a divisor of t, then 3 is in the prime factorization of t
Since the prime factorization of 12 also has a 3 in it, we can see that we can now answer the reworded target question.
As such, statement 1 is sufficient.
Statement 2: m is a multiple of 4.
In other words,
4 is hiding in the prime factorization of m
So, we know that m =
(2)(2)(?)(?)(?)...
From here, we'll take our given information, 2k + 3m = t and replace m with
(2)(2)(?)(?)(?) to get: 2k + 3
(2)(2)(?)(?)(?) = t
At this point, we can factor out a 2 to get:
2[k + 3(2)(?)(?)(?) = t, which means 2 is a divisor of t
If 2 is a divisor of t, then 2 is in the prime factorization of t
Since the prime factorization of 12 also has a 2 in it, we can see that we can now answer the reworded target question.
As such, statement 2 is sufficient, and the answer is
D
Cheers,
Brent
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