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# Is the positive integer X divisible by 21?

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Math Expert
Joined: 02 Sep 2009
Posts: 57191
Is the positive integer X divisible by 21?  [#permalink]

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14 Mar 2016, 07:47
10
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Difficulty:

75% (hard)

Question Stats:

53% (01:51) correct 47% (02:18) wrong based on 129 sessions

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Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

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Is the positive integer X divisible by 21?  [#permalink]

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16 Mar 2016, 05:58

1st : 14n + 4

2st : 15n + 5

plugged in few numbers, and checked whether x is divisible by 21 or not.
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Re: Is the positive integer X divisible by 21?  [#permalink]

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16 Mar 2016, 07:45
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

statement 1 may be yes or no.
statement 2 may be yes or no.
But combining two statements you can see that from statement 1, X is divisible by 7 and from statement 2, X is divisible by 3. So X wiil be divisible by 21.

................
thanks
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Re: Is the positive integer X divisible by 21?  [#permalink]

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16 Mar 2016, 08:00
3
KanonKumarSen wrote:
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

statement 1 may be yes or no.
statement 2 may be yes or no.
But combining two statements you can see that from statement 1, X is divisible by 7 and from statement 2, X is divisible by 3. So X wiil be divisible by 21.

................
thanks

Hi,

If you look at the choices and read into the statements you will find you have enough information for the answer..

For the positive integer X to be divisible by 21, X should be a multiple of 3 and 7..

lets see the statements --

(1) When X is divided by 14, the remainder is 4
If X leaves a remainder 0f 4 when divided by 14, it will leave a remainder 4 when div by 7..
so its not div by 7 and thus our answer will always be NO
Suff

(2) When X is divided by 15, the remainder is 5
If X leaves a remainder 0f 5 when divided by 15, it will leave a remainder 5 or 5-3=2 when div by 3..
so its not div by 3 and thus our answer will always be NO
Suff

D

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Re: Is the positive integer X divisible by 21?  [#permalink]

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22 Mar 2016, 02:35
1
Nice question here the logic to see that in order to check the divisibility by 21 => number should be both divisible by 7 and 3
and we can conclude the D is the answer number would never be divisible by 21
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Re: Is the positive integer X divisible by 21?  [#permalink]

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22 Mar 2016, 23:56
chetan2u wrote:
KanonKumarSen wrote:
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

statement 1 may be yes or no.
statement 2 may be yes or no.
But combining two statements you can see that from statement 1, X is divisible by 7 and from statement 2, X is divisible by 3. So X wiil be divisible by 21.

................
thanks

Hi,

If you look at the choices and read into the statements you will find you have enough information for the answer..

For the positive integer X to be divisible by 21, X should be a multiple of 3 and 7..

lets see the statements --

(1) When X is divided by 14, the remainder is 4
If X leaves a remainder 0f 4 when divided by 14, it will leave a remainder 4 when div by 7..
so its not div by 7 and thus our answer will always be NO
Suff

(2) When X is divided by 15, the remainder is 5
If X leaves a remainder 0f 5 when divided by 15, it will leave a remainder 5 or 5-3=2 when div by 3..
so its not div by 3 and thus our answer will always be NO
Suff

D

oh! Now i understand.
thanks
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Joined: 12 Sep 2015
Posts: 3912
Re: Is the positive integer X divisible by 21?  [#permalink]

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25 Mar 2017, 06:36
1
Top Contributor
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

Target question: Is the positive integer x divisible by 21?

Statement 1: When X is divided by 14, the remainder is 4
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

In this statement, we aren't told how many times 14 divides into x, but we can just assume that 14 divides into x k times. In other words, x divided by 14 equals k with remainder 4"
So, we can write: x = 14k + 4 (where k is some positive integer)
This is enough information to show that x is definitely NOT divisible by 21
How do I know this?

Take x = 14k + 4 and rewrite 14 as follows: x = (7)(2)k + 4
Highlight two values: x = (7)(2)k + 4
Simplify: x = (7)(some integer) + 4
This tells us that x is 4 GREATER than some multiple of 7
In other words, x is NOT divisible by 7.
If x is NOT divisible by 7, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When X is divided by 15, the remainder is 5
Let's say that 15 divides into x j times. In other words, x divided by 15 equals j with remainder 5"
So, we can write: x = 15j + 5 (where j is some positive integer)
Rewrite the equation: x = (3)(5)j + 5
Highlight two values: x = (3)(5)k + 5
Simplify: x = (3)(some integer) + 5
This tells us that x is 5 GREATER than some multiple of 3
In other words, x is NOT divisible by 3.
If x is NOT divisible by 3, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

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Re: Is the positive integer X divisible by 21?  [#permalink]

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25 Mar 2017, 11:45
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

Here,
Statement 1 says:
x=18,32,46,60,74 and so on like that
Statement 2 says:
x=20,35,50,65,80 and so on like that
So, the correct choice is
Thank you...
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Re: Is the positive integer X divisible by 21?  [#permalink]

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16 Aug 2019, 10:27
GMATPrepNow wrote:
Bunuel wrote:
Is the positive integer X divisible by 21?

(1) When X is divided by 14, the remainder is 4
(2) When X is divided by 15, the remainder is 5

Target question: Is the positive integer x divisible by 21?

Statement 1: When X is divided by 14, the remainder is 4
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

In this statement, we aren't told how many times 14 divides into x, but we can just assume that 14 divides into x k times. In other words, x divided by 14 equals k with remainder 4"
So, we can write: x = 14k + 4 (where k is some positive integer)
This is enough information to show that x is definitely NOT divisible by 21
How do I know this?

Take x = 14k + 4 and rewrite 14 as follows: x = (7)(2)k + 4
Highlight two values: x = (7)(2)k + 4
Simplify: x = (7)(some integer) + 4
This tells us that x is 4 GREATER than some multiple of 7
In other words, x is NOT divisible by 7.
If x is NOT divisible by 7, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: When X is divided by 15, the remainder is 5
Let's say that 15 divides into x j times. In other words, x divided by 15 equals j with remainder 5"
So, we can write: x = 15j + 5 (where j is some positive integer)
Rewrite the equation: x = (3)(5)j + 5
Highlight two values: x = (3)(5)k + 5
Simplify: x = (3)(some integer) + 5
This tells us that x is 5 GREATER than some multiple of 3
In other words, x is NOT divisible by 3.
If x is NOT divisible by 3, then we can be certain that x is NOT divisible by 21.
We know this because, for x to be divisible by 21, x must be divisible by 3 AND x must be divisible by 7
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

RELATED VIDEO FROM OUR COURSE

Do we really need to go into this much depth when thinking about the problem?

Surely just by looking at X = 14q + 4 you can see that there is a remainder of 14 if you do 14q/21 and a remainder of 4 if you do 4/21. So that tells you it doesn't divide and you don't need to probe deeper? Same logic for the second statement..
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Is the positive integer X divisible by 21?  [#permalink]

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16 Aug 2019, 10:38
Top Contributor
jamalabdullah100 wrote:
Surely just by looking at X = 14q + 4 you can see that there is a remainder of 14 if you do 14q/21 and a remainder of 4 if you do 4/21. So that tells you it doesn't divide and you don't need to probe deeper? Same logic for the second statement..

I'm not sure everyone would agree that the solution is as obvious as you suggest, especially since only 55% of students answered it correctly.
That said, I may be guilty of over-explaining.

Cheers,
Brent
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Re: Is the positive integer X divisible by 21?  [#permalink]

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16 Aug 2019, 10:44
GMATPrepNow wrote:
jamalabdullah100 wrote:
Surely just by looking at X = 14q + 4 you can see that there is a remainder of 14 if you do 14q/21 and a remainder of 4 if you do 4/21. So that tells you it doesn't divide and you don't need to probe deeper? Same logic for the second statement..

I'm not sure everyone would agree that the solution is as obvious as you suggest, especially since only 55% of students answered it correctly.
That said, I may be guilty of over-explaining.

Cheers,
Brent

I was just double checking that I wasn't over-simplifying and lucky to get the answer. Thanks!
Re: Is the positive integer X divisible by 21?   [#permalink] 16 Aug 2019, 10:44
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