If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?
(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
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what is the remainder when the product abc is divided by 2?
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Hi rsarashi,
This question is a great 'concept' question built around a few Number Property rules (meaning that if you understand the concepts involved, you don't have to do much math to get to the correct answer).
We're told that A, B and C are all positive integers that are each greater than 1. We're asked for the remainder when (A)(B)(C) is divided by 2. Since this prompt is based around the idea of dividing numbers by 2, there's a Number Property rule worth noting: when dividing by 2, the remainder can only be two possible numbers: 0 or 1. Notice the pattern....
1/2 = 0 r 1
2/2 = 1 r 0
3/2 = 1 r 1
4/2 = 2 r 0
Etc.
Notice that EVEN numbers have a remainder of 0 and ODD numbers have a remainder of 1.
1) If each of A, B, and C is divided by 2, the product of all three remainders is 0.
Fact 1 tells us that the PRODUCT of the three remainders is 0, which means that at least one of the remainders is 0. By extension, this tells us that at least one of the three numbers is EVEN. When multiplying integers, if one of the integers is EVEN, then the product will ALWAYS be EVEN. Thus, since (A)(B)(C) will be EVEN, dividing that product by 2 will ALWAYS give us a remainder of 0.
Fact 1 is SUFFICIENT.
(2) If each of A, B, and C is divided by 2, the SUM of all three remainders is 2.
Since the remainders can only be 0 or 1, if the sum of the three remainders is 2, then two of the numbers are ODD and the third number is EVEN. We have a similar situation to the one that we discussed in Fact 1 above: (A)(B)(C) will be EVEN, and the remainder of that product will always be 0.
Fact 1 is SUFFICIENT.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
This question is a great 'concept' question built around a few Number Property rules (meaning that if you understand the concepts involved, you don't have to do much math to get to the correct answer).
We're told that A, B and C are all positive integers that are each greater than 1. We're asked for the remainder when (A)(B)(C) is divided by 2. Since this prompt is based around the idea of dividing numbers by 2, there's a Number Property rule worth noting: when dividing by 2, the remainder can only be two possible numbers: 0 or 1. Notice the pattern....
1/2 = 0 r 1
2/2 = 1 r 0
3/2 = 1 r 1
4/2 = 2 r 0
Etc.
Notice that EVEN numbers have a remainder of 0 and ODD numbers have a remainder of 1.
1) If each of A, B, and C is divided by 2, the product of all three remainders is 0.
Fact 1 tells us that the PRODUCT of the three remainders is 0, which means that at least one of the remainders is 0. By extension, this tells us that at least one of the three numbers is EVEN. When multiplying integers, if one of the integers is EVEN, then the product will ALWAYS be EVEN. Thus, since (A)(B)(C) will be EVEN, dividing that product by 2 will ALWAYS give us a remainder of 0.
Fact 1 is SUFFICIENT.
(2) If each of A, B, and C is divided by 2, the SUM of all three remainders is 2.
Since the remainders can only be 0 or 1, if the sum of the three remainders is 2, then two of the numbers are ODD and the third number is EVEN. We have a similar situation to the one that we discussed in Fact 1 above: (A)(B)(C) will be EVEN, and the remainder of that product will always be 0.
Fact 1 is SUFFICIENT.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
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Dividing an EVEN NUMBER by 2 yields a remainder of 0.rsarashi wrote:If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?
(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
Dividing an ODD NUMBER by 2 yields a remainder of 1.
Thus, to determine the remainder when abc is divided by 2, we need to know whether abc is even or odd.
abc = odd only if a, b, and c are all odd.
Question stem, rephrased:
Are a, b and c all odd?
Statement 1:
For the product of the three remainders to be 0, at least one of the three remainders must be 0.
Since only an even integer will yield a remainder of 0 when divided by 2, at least one of the three integers must be EVEN.
Thus, a, b, and c are NOT all odd, and the answer to the rephrased question stem is NO.
SUFFICIENT.
Statement 2:
If a, b and c are all odd, then each will yield a remainder of 1 when divided by 2, with the result that the sum of the three remainders = 1+1+1 = 3.
Since the sum of the three remainder is not 3, it is not possible that a, b and c are all odd.
Thus, the answer to the rephrased question stem is NO.
SUFFICIENT.
The correct answer is D.
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
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ceilidh.erickson
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The question "what is the remainder when the product abc is divided by 2?" is a great example of CODED LANGUAGE on the GMAT. "Remainder when divided by 2" is just a convoluted way of testing EVENS / ODDS, as the other experts have demonstrated.
Here's an article that can help you to practice translating the GMAT's coded language into plain English: https://www.manhattanprep.com/gmat/blog ... uage-test/
Here's an article that can help you to practice translating the GMAT's coded language into plain English: https://www.manhattanprep.com/gmat/blog ... uage-test/
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
