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If x, y, and d are integers and d is odd, are both x and y

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If x, y, and d are integers and d is odd, are both x and y

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Source: GMAT Paper Tests

If x, y, and d are integers and d is odd, are both x and y divisible by d?

1) x+y is divisible by d.
2) x-y is divisible by d.

The OA is C

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BTGmoderatorLU wrote:
Source: GMAT Paper Tests

If x, y, and d are integers and d is odd, are both x and y divisible by d?

1) x+y is divisible by d.
2) x-y is divisible by d.

The OA is C
Given: x, y, and d are integers and d is odd

We have to find out whether x, y, and d are integers and d is odd.

Let's take each statement one by one.

1) x + y is divisible by d.

Case 1: Say x = 5, y = 1 and d =3

x + y = 5 + 1 = 6; we see that 6 is divisible 3, but neither x nor y is divisible by d. The answer is No.

Case 2: Say x = 6, y = 3 and d =3

x + y = 6 + 3 = 9; we see that 9 is divisible 3, and both x are y divisible by d. The answer is Yes.

Insufficient.

2) x - y is divisible by d.

Case 1: Say x = 5, y = 2 and d =3

x - y = 5 - 2 = 3; we see that 3 is divisible 3, but neither x nor y is divisible by d. The answer is No.

Case 2: Say x = 6, y = 3 and d =3

x - y = 6 - 3 = 3; we see that 3 is divisible 3, and both x are y divisible by d. The answer is Yes.

Insufficient.

(1) and (2) together

We see that Case 1 of both the statements cannot exist here; thus, the answer is yes, both x and y are divisible by d.

The correct answer: C

Hope this helps!

-Jay
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BTGmoderatorLU wrote:
Source: GMAT Paper Tests

If x, y, and d are integers and d is odd, are both x and y divisible by d?

1) x+y is divisible by d.
2) x-y is divisible by d.

The OA is C
Given: x, y, and d are integers and d is odd

Target question: Are both x and y divisible by d?

Statement 1: x+y is divisible by d.
Let's TEST some values.
There are several values of x, y and d that satisfy statement 1. Here are two:
Case a: x = 6, y = 9 and d = 3. Notice that 6+9 = 15, and 15 is divisible by 3. In this case, the answer to the target question is YES, x and y ARE both divisible by d
Case b: x = 2, y = 4 and d = 3. Notice that 2+4 = 6, and 6 is divisible by 3. In this case, the answer to the target question is NO, x and y are NOT both divisible by d
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x-y is divisible by d
Let's TEST some values.
Case a: x = 9, y = 3 and d = 3. Notice that 9-3 = 6, and 6 is divisible by 3. In this case, the answer to the target question is YES, x and y ARE both divisible by d
Case b: x = 10, y = 4 and d = 3. Notice that 10-4 = 6, and 6 is divisible by 3. In this case, the answer to the target question is NO, x and y are NOT both divisible by d
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Here are some nice divisibility rules:
Rule #1. If integers A and B are each divisible by integer k, then (A + B) is divisible by k
Rule #2. If integers A and B are each divisible by integer k, then (A - B) is divisible by k


Statement 1 tells us that x+y is divisible by d
Statement 2 tells us that x-y is divisible by d
By Rule #1, (x+y) + (x-y) is divisible by d
Simplify to get: 2x is divisible by d
Since we're told that d is ODD, we know that x MUST BE divisible by d

By Rule #2, (x+y) - (x-y) is divisible by d
Simplify to get: 2y is divisible by d
Since we're told that d is ODD, we know that y MUST BE divisible by d

So, the answer to the target question is YES, x and y ARE both divisible by d

Answer: C

Cheers,
Brent

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Question --> are both x and y divisible by d?
This means that x and y must be a multiple of d

STATEMENT 1: x+y is divisible by d
$$Let\ \ d\ =3,\ and\ x+y=21$$ #
$$Case\ 1:\ If\ x=6\ and\ y=15,\ then\ x\ and\ y\ are\ divisible\ by\ d$$
$$Case\ 2:\ But\ if\ x=1\ and\ y=20,\ both\ x\ and\ u\ are\ not\ divisible\ by\ d$$
$$Therefore,\ statement\ 1\ is\ INSUFFICIENT$$

STATEMENT 2: x-y is divisible by d
let d=3 and x-y=21
$$Case\ 1:\ if\ x=25\ and\ y=4,\ then\ x\ and\ y\ are\ not\ divisible\ by\ d$$
$$Case\ 2:\ if\ x=24\ and\ y=3,\ then\ x\ and\ y\ are\ divisible\ by\ d$$
$$Thus,\ statement\ 2\ is\ INSUFFICIENT$$

Combining both statement together;
Statement 1 = (x+y)/3 = v ------ (1)
Statement 2 = (x-y)/3 = w -------(2)
$$From\ \left(2\right),\ x=3w+y$$
$$From\ \left(1\right),\ x+y=3v\ where\ x=3w+y$$
$$3w+y+y=3v$$
$$2y=3\left(v-w\right)\ and\ hence,\ y=\frac{3\left(v-w\right)}{2}$$
Since v and w are whole numbers / integer, then y must be a multiple of 3.
x=3w + y where y is a multiple of 3 and x also must be a multiple of 3.
Both x and y are divisible by d when both statement are combined together.

Therefore, option C is the correct answer

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