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If x and y are positive integers, what is the remainder when

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If x and y are positive integers, what is the remainder when

by Max@Math Revolution » Wed May 02, 2018 1:52 am

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[GMAT math practice question]

If x and y are positive integers, what is the remainder when x+y is divided by 2?

1) xy is divisible by 4
2) y is divisible by 2
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed May 02, 2018 5:33 am

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Max@Math Revolution wrote:[GMAT math practice question]

If x and y are positive integers, what is the remainder when x+y is divided by 2?

1) xy is divisible by 4
2) y is divisible by 2
Target question: What is the remainder when x+y is divided by 2?

Statement 1: xy is divisible by 4
Let's TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 2 and y = 4. Here xy = (2)(4) = 8, and 8 is divisible by 4. In this case, x +y = 2 + 4 = 6, and when we divide 6 by 2 the remainder is 0. So, the answer to the target question is 0
Case b: x = 3 and y = 4. Here xy = (3)(4) = 12, and 12 is divisible by 4. In this case, x +y = 3 + 4 = 7, and when we divide 7 by 2 the remainder is 1. So, the answer to the target question is 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y is divisible by 2
Since we have no information about the value of x, we cannot answer the target question with certainty.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of x and y that satisfy BOTH statements. Here are two:
Case a: x = 2 and y = 4. Here xy = (2)(4) = 8, and 8 is divisible by 4. In this case, x +y = 2 + 4 = 6, and when we divide 6 by 2 the remainder is 0. So, the answer to the target question is 0
Case b: x = 3 and y = 4. Here xy = (3)(4) = 12, and 12 is divisible by 4. In this case, x +y = 3 + 4 = 7, and when we divide 7 by 2 the remainder is 1. So, the answer to the target question is 1
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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by Jeff@TargetTestPrep » Thu May 03, 2018 3:32 pm

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Max@Math Revolution wrote:[GMAT math practice question]

If x and y are positive integers, what is the remainder when x+y is divided by 2?

1) xy is divisible by 4
2) y is divisible by 2
We need to determine the remainder of x + y divided by 2, or, in other words, we need to determine whether x + y is even or odd.

Statement One Alone:

xy is divisible by 4

We see that xy is a multiple of 4. If xy is 8, we could have x = 2 and y = 4, so x + y is even, or we could have x = 1 and y = 8, in which case x + y would be odd. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

y is divisible by 2

Since we have no information regarding x, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using our two statements together, we still cannot answer the question. We could have x = 2 and y = 4, so x + y is even, or we could have x = 1 and y = 8, in which case x + y would be odd.

Answer: E

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by Max@Math Revolution » Thu May 03, 2018 11:32 pm

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):
If x = 4 and y = 1, x + y = 5 has remainder 1 when it is divided by 4.
If x = 2 and y = 2, x + y = 4 has remainder 0 when it is divided by 4.

Since we don't have a unique solution, both conditions together are not sufficient.

Therefore, E is the answer.

Answer: E

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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by deloitte247 » Mon May 07, 2018 12:06 pm

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$$\frac{\left(x+y\right)}{2}=\ T\arg et\ question$$
If two positive integers x and y is divisible by 2 the only position remainder is 0 and 1
1st statement x and y is divisible by 4 case study to test value
if x= 2 and y= 4 : xy = 2*4 = 8
8 is divisible by 4 implementing the value into $$\frac{x+y}{2}\ =\ \frac{2+4}{2}\ =\ \frac{6}{2\ }=\ 3.0$$
if x= 3 and y = 4 xy = 3*4 = 12
12 is divisible by 4 implementing the value into
$$\frac{\left(x+y\right)}{2}=\ \frac{\left(3+4\right)}{2}=\frac{7}{2}=3r1$$
This certifies that $$\frac{\left(x+y\right)}{2}is\ not\ definite\ and\ cannot\ be\ answered\ with\ certainty$$
therefore the first statement is not SUFFICIENT
2nd statement , therefore Y is divisible by 2
No information is given about integer x so the answer to the question is not definite
Therefore, the 2nd statement is NOT SUFFICIENT
Statement (1) and (2) together are NOT SUFFICIENT.
Hence, option E is the answer
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