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# If a, b, and c are three positive integers, each greater

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Manager
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If a, b, and c are three positive integers, each greater  [#permalink]

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03 Aug 2014, 08:17
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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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Re: If a, b, and c are three positive integers, each greater  [#permalink]

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03 Aug 2014, 09:40
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oss198 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.

For this, you need to have some understand of number properties related to odds/evens. Here are the "rules" to know which would help you answer the question:

1) Odd x Odd = Odd
2) Odd x Even = Even
3) Even x Even = Even
4) Odd / 2 = remainder 1
5) Even / 2 = remainder 0

The question is essentially asking if the product of a, b, c is even or odd.

So now let's look at each statement.

Statement 1: If each of a,b,c is divided by 2 the product of the remainders is 0. According to rules number 4 and 5, if a, b, and c are all odd, the product of the remainders would be 1. So one of them must be even. Then, if you know that at least one of the numbers is even, then according to rules 1, 2, and 3 the product of a, b, c must be even. Therefore this statement is sufficient.

We can now eliminate B, C, and E.

Statement 2: If the sum of the remainders is 2, that means of a, b, c, two of them must be odd and one is even. Now we're in the same situation as the previous statement of having at least 1 even number, therefor the remainder of product of a, b, c must be zero. This statement is also sufficient.
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Re: If a, b, and c are three positive integers, each greater  [#permalink]

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03 Aug 2014, 23:32
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oss198 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.

(1) if the product of all three remainders is 0, then at least one of the three a ,b,c is divisible by 2. for instance, a is even, can be divided by 2, then the product abc is also divisible by 2
(2), if the sum of all three remainders is 2 , then each remainder will be 0,1,1. At least one of the three a ,b,c is divisible by 2, the product abc is also divisible by 2

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Re: If a, b, and c are three positive integers, each greater  [#permalink]

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22 May 2017, 20:36
oss198 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.

Here this problem can be solved easily via some intuition. The problem asks what the remainder is when ABC/2. In other words, is ABC even? If so, then the remainder is 0 and the info is sufficient.

GOAL: What is the remainder (unique, distinct value) of the product of A, B, and C when that quantity is divided by 2?

Statement 1: We know that when the remainders of A, B, and C are multiplied together, the result is 0. We thus know that at least one of the number is even since its remainder when divided by 2 is 0. So that means that the quantity A*B*C is even and thus the remainder of ABC/2 = 0. Sufficient.

Statement 2: We know that the sum of the remainders = 2. So that means that two of the remainders = 1 and the other = 0. None of the remainders can be negative or greater than or equal to 2. So there is only one possible set of values: {0,1,1}. In this case, one of the values is even so that means the quantity A*B*C is even and thus A*B*C/2 remainder is 0. Sufficient.
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Re: If a, b, and c are three positive integers, each greater  [#permalink]

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19 Jul 2017, 20:46
oss198 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.

Beautiful Question

They key concept to this question is that if either a,b or c is a multiple of 2- in other words if at least any one of these variables has a remainder of 0 then the product of all 3 will have no remainder because it would be a multiple of 2.

St 1

This implies that at least one of the integers a, b or c has a remainder of 0- so that means at least one of these integers a,b or c is a multiple of 2. Sufficient

St 2

This implies that either 2 of the integers have a remainder of 1 and the other a remainder of 0, or that two of the integers have no remainder and one has a remainder of 2. In either case, because at least one of the integers is a multiple of 2 the product must be a multiple of 2- or in simpler terms divisible of 2. Sufficient

D
Re: If a, b, and c are three positive integers, each greater   [#permalink] 19 Jul 2017, 20:46
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