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# If x and y are positive integers, is x divisible by 3?

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Joined: 29 Jan 2015
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If x and y are positive integers, is x divisible by 3?  [#permalink]

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22 Jul 2019, 00:35
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If x and y are positive integers, is x divisible by 3?

(1) x-y is divisible by 3.

(2) x-2y is divisible by 3.

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If x and y are positive integers, is x divisible by 3?  [#permalink]

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Updated on: 23 Jul 2019, 00:56
rohan2345 wrote:
If x and y are positive integers, is x divisible by 3?

(1) x-y is divisible by 3.

(2) x-2y is divisible by 3.

Given: x and y are positive integers.
Asked: Is x divisible by 3?

(1) x-y is divisible by 3.
x-y = 3k where k is an integer=> x-y =3,6,9.... If y =1 x=4,7,... but if y=3, x=3,6,9...
NOT SUFFICIENT

(2) x-2y is divisible by 3.
x-2y=3k where k is an integer => x-2y =3,6,9.... If y=1 x=5,8,... but if y=3, x=3,6,9...
NOT SUFFICIENT

Combining (1) & (2)
(1) x-y is divisible by 3.
(2) x-2y is divisible by 3.
Subtracting (2) from (1)
(x-y)-(x-2y) = y is divisible by 3
From (1) x is divisible by 3.
SUFFICIENT

IMO C
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Originally posted by Kinshook on 23 Jul 2019, 00:29.
Last edited by Kinshook on 23 Jul 2019, 00:56, edited 1 time in total.
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Re: If x and y are positive integers, is x divisible by 3?  [#permalink]

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23 Jul 2019, 00:30
(1) $$x-y$$ is divisible by 3.

If $$x=5$$, $$y=2$$ - NO
If $$x=6$$, $$y=3$$ - YES

1 is insufficient

(2) $$x-2y$$ is divisible by 3.

If $$x=5$$, $$y=1$$ - NO
If $$x=12$$, $$y=3$$ - YES

2 is insufficient

(1) + (2)

From (1) $$x-y = 3n$$

Subtracting both sides by $$y$$

$$x-2y = 3n - y$$

Since from (2) $$x-2y$$ is divisible by 3, $$3n - y$$ is divisible by 3

Since $$3n$$ is divisible by 3 and $$3n-y$$ is divisible by 3, $$y$$ must be divisible by 3

If $$y$$ is divisible by 3 and $$x-y$$ is divisible by 3, then $$x$$ must be divisible by 3

Sufficient

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Re: If x and y are positive integers, is x divisible by 3?  [#permalink]

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23 Jul 2019, 02:46
Statement1: x-y is divisible by 3
a) 6-3=3 (yes, 6 is divisible by 3)
b) 7-4=3 (no, 7 is not divisible by 3)
Insufficient

Statement2: x-2y is divisible by 3
a) 15-2*3=9 ( yes, 15 is divisible by 3)
b) 14-2*4=6 (no, 14 is not divisible by 3)
Insufficient.

Taken together 1 and 2,
Let’s say that
—> x-y=3*a (a is quotient)
And
—> x-2y=3*b (b is quotient)

Subtract small one from bigger one:
(x-y)-(x-2y)=3*a-3*b=3(a-b)
y=3(a-b)
that means y is divisible by 3

As statement mentioned, x-y is divisible by 3–> if y is divisible by 3, x must be divisible by 3.
Sufficient

The answer choice is C.

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Re: If x and y are positive integers, is x divisible by 3?  [#permalink]

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23 Jul 2019, 04:19
rohan2345 wrote:
If x and y are positive integers, is x divisible by 3?

(1) x-y is divisible by 3.

(2) x-2y is divisible by 3.

If you understand the concept of divisibility and grouping, this can be done in a few seconds.

(1) x-y is divisible by 3.

Just because the difference is divisible doesn't mean the numbers are multiples of 3 too. e.g. 8-5 = 3.

(2) x-2y is divisible by 3.

Same as above. e.g. x = 8 and y = 1 so x - 2y = 8 - 2 = 6

Using both, when we subtract y out of x, we are left with a multiple of 3. When we subtract another y, we are again left with a multiple of 3. This means y must be a multiple of 3 too since you must have taken out groups of 3 when you took out y. Then x MUST be a multiple of 3 too since x is made by adding a multiple of 3 to y which is a multiple of 3 too.

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Re: If x and y are positive integers, is x divisible by 3?   [#permalink] 23 Jul 2019, 04:19
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# If x and y are positive integers, is x divisible by 3?

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