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# Is integer x divisible by 21? (1) x is divisible by 14.

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Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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24 Mar 2014, 03:45
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84% (00:35) correct 16% (00:21) wrong based on 122 sessions

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Is integer x divisible by 21?

(1) x is divisible by 14.
(2) x is divisible by 12.

OE:
[Reveal] Spoiler:
(1): x can be 14, which is indivisible by 21, or by 14*21, in which case it is divisible by 21.
Insufficient
(2): x can be 12, which is indivisible by 21, or by 12*21, in which case it is divisible by 21.
Insufficient
Combined: x is divisible by 14 and by 12
x must be divisible by the prime factors of 14=2*7 and by those of 12=2*2*3, thus x is divisible by 7 and by 3
-> x is divisible by 7*3=21.
Sufficient
[Reveal] Spoiler: OA

Kudos [?]: 1790 [1], given: 289

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Re: Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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24 Mar 2014, 03:57
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goodyear2013 wrote:
Is integer x divisible by 21?

(1) x is divisible by 14.
(2) x is divisible by 12.

OE:
[Reveal] Spoiler:
(1): x can be 14, which is indivisible by 21, or by 14*21, in which case it is divisible by 21.
Insufficient
(2): x can be 12, which is indivisible by 21, or by 12*21, in which case it is divisible by 21.
Insufficient
Combined: x is divisible by 14 and by 12
x must be divisible by the prime factors of 14=2*7 and by those of 12=2*2*3, thus x is divisible by 7 and by 3
-> x is divisible by 7*3=21.
Sufficient

Is integer x divisible by 21?

(1) x is divisible by 14 --> if x=42, then YES but if x=14, then NO. Not sufficient.

(2) x is divisible by 12 --> if x=84, then YES but if x=12, then NO. Not sufficient.

(1)+(2) From (1) x is divisible by 7 and from (2) x is divisible by 3, thus x must be divisible by the least common multiple of 7 and 3, which is 21. Sufficient.

Hope it's clear.
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Re: Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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23 Jul 2014, 21:18
goodyear2013 wrote:
Is integer x divisible by 21?

(1) x is divisible by 14.
(2) x is divisible by 12.

OE:
[Reveal] Spoiler:
(1): x can be 14, which is indivisible by 21, or by 14*21, in which case it is divisible by 21.
Insufficient
(2): x can be 12, which is indivisible by 21, or by 12*21, in which case it is divisible by 21.
Insufficient
Combined: x is divisible by 14 and by 12
x must be divisible by the prime factors of 14=2*7 and by those of 12=2*2*3, thus x is divisible by 7 and by 3
-> x is divisible by 7*3=21.
Sufficient

I just want to try to explain my thinking (a little different method than Bunuel) so in case it's helpful to others..
Question asks: is x divisible by 21.. I factored out 21 = 7*3
s1. x is divisible by 14 14 = 7*2 --> not sufficient no 3 here.
s2. x is divisible by 12 12= 2*2*3 --> no 7 here so insuff.
s1+s2 we have 2,3,7 hence x is divisible by 21 (C).

I'm very new here and trying to get better with math. using this to express my method (also in case if it's wrong somebody can correct me) and hope it help to other people.
Cheers

Kudos [?]: [0], given: 10

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Joined: 27 Aug 2012
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Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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23 Jul 2014, 21:59
prime factors of 21 are 3,7
So to x be divisible 21 it should have 3 and 7 in its prime box
Prime factors of 14(2,7)
so statement 1 is not sufficient, it shows that there is 2,7 present in the prime box of x, but we dont know what x is, So we are not sure if we have 3 present in the prime box of X

prime factors of 12 (2,2,3)
statement 2 is not sufficient, because we dont know if 7 is present in the prime box of x

now consider both the statements together
x is having following factors in its prime box(2,2,7,3) or (2,2,2,7,3) because one of the 2 could be common or different.
So from the above statemnts together we can say that x is divisible by 21, because (3,7) are present in its prime box.

Cheers,
Navjot

Kudos [?]: 4 [0], given: 4

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Re: Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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17 Jul 2017, 20:24
goodyear2013 wrote:
Is integer x divisible by 21?

(1) x is divisible by 14.
(2) x is divisible by 12.

OE:
[Reveal] Spoiler:
(1): x can be 14, which is indivisible by 21, or by 14*21, in which case it is divisible by 21.
Insufficient
(2): x can be 12, which is indivisible by 21, or by 12*21, in which case it is divisible by 21.
Insufficient
Combined: x is divisible by 14 and by 12
x must be divisible by the prime factors of 14=2*7 and by those of 12=2*2*3, thus x is divisible by 7 and by 3
-> x is divisible by 7*3=21.
Sufficient

We can use properties of LCM and prime factorization to solve

Statement 1
x= 7 x 2 x some integer K

insufficient because X could be a small number like 14

Statement 2

x= 3 x 2 x 2 x some integer K

insufficient x could be some small number like 24

Statement 1 & 2

We should find the LCM of both numbers - which can be done through prime factorization

LCM (14, 12) = 7 x 3 x 2 x 2

https://www.khanacademy.org/math/algebr ... e-exercise - he does GMAT stuff now lol

C

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Re: Is integer x divisible by 21? (1) x is divisible by 14. [#permalink]

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30 Jul 2017, 17:39
goodyear2013 wrote:
Is integer x divisible by 21?

(1) x is divisible by 14.
(2) x is divisible by 12.

We need to determine whether x/21 = integer

Statement One Alone:

x is divisible by 14.

Statement one alone is not sufficient to answer the question. If x = 14, then x/21 is not an integer; however, if x = 42, then x/21 is an integer.

Statement Two Alone:

x is divisible by 12.

Statement two alone is not sufficient to answer the question. If x = 12, then x/21 is not an integer; however, if x = 84, then x/21 is an integer.

Statements One and Two Together:

Using statements one and two, we see that x is a multiple of 14 and 12, and thus it is a multiple of their least common multiple, which is 84. Since all multiples of 84 are divisible by 21, x/21 = integer.

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Re: Is integer x divisible by 21? (1) x is divisible by 14.   [#permalink] 30 Jul 2017, 17:39
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