-Patrick
Is xy a multiple of 105?
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Source: Beat The GMAT — Data Sufficiency |
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Patrick_GMATFix
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When a question asks about factors, multiples or primes, it is usually a good idea to break your numbers into their prime factors. The answer is B. I go through the question in detail in the full solution below (taken from the GMATFix App).
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Brent@GMATPrepNow
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Target question: Is xy a multiple of 105?GmatGreen wrote:If positive integer x is a multiple of 6 and positive integer y is a multiple of 14, is xy a multiple of 105?
1) x is a multiple of 9
2) y is a multiple of 25
Important stuff:
First, If N is a multiple of k, then N is divisible by k.
Second, 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
Examples:
24 is divisible by 3 <--> 24 = 2x2x2x3
70 is divisible by 5 <--> 70 = 2x5x7
330 is divisible by 6 <--> 330 = 2x3x5x11
56 is divisible by 8 <--> 56 = 2x2x2x7
Since 105 = (3)(5)(7), then we can rewrite the target question as . . .
Rephrased target question: Is there a 3, a 5 and a 7 hiding in the prime factorization of xy?
Given: x is a multiple of 6
In other words, x = (2)(3)(other possible prime numbers)
Given: y is a multiple of 14
In other words,y = (2)(7)(other possible prime numbers)
Combine both of the above to see that xy = (2)(2)(3)(7)(other possible prime numbers)
So, the given information tells us that we ALREADY have a 3 and a 7 hiding in the prime factorization of xy. The only piece missing is the 5.
So, we can rephrase our target question one last time. . .
Rephrased target question: Is there a 5 hiding in the prime factorization of xy?
Now we can check the statements.
Statement 1: x is a multiple of 9.
Since 9 = (3)(3), all this tells us is that there are two 3's hiding in the prime factorization of xy.
So, there may or may not be a 5 hiding in the prime factorization of xy.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: y is a multiple of 25.
Since 25 = (5)(5), this tells us is that there is definitely a 5 hiding in the prime factorization of xy.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
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Brent
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ceilidh.erickson
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If x is a multiple of 6, then we know it contains prime factors of 2 and 3. If y is a multiple of 14, then we know it contains prime factors of 2 and 7. We therefore know that the product xy will contain all of the prime factors that x and y have in them respectively: 2, 2, 3, and 7.
The question asks: is xy a multiple of 105?
To be a multiple of 105, a number would have to contain all of the prime factors of 105. So first, break it down into primes. 105 = 3*5*7
We already know that xy has 3 and a 7 as factors. That means that our target question is: does xy have a factor of 5? In other words, does either x or y have a factor of 5?
(1) x is a multiple of 9
This doesn't tell us if it had a factor of 5. Insufficient.
(2) y is a multiple of 25
If it's a multiple of 25, then it has 2 factors of 5. Sufficient!
The answer is B.
The question asks: is xy a multiple of 105?
To be a multiple of 105, a number would have to contain all of the prime factors of 105. So first, break it down into primes. 105 = 3*5*7
We already know that xy has 3 and a 7 as factors. That means that our target question is: does xy have a factor of 5? In other words, does either x or y have a factor of 5?
(1) x is a multiple of 9
This doesn't tell us if it had a factor of 5. Insufficient.
(2) y is a multiple of 25
If it's a multiple of 25, then it has 2 factors of 5. Sufficient!
The answer is B.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
