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if the least common multiple of positive integer z and 24 is 72, and G

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if the least common multiple of positive integer z and 24 is 72, and G  [#permalink]

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New post Updated on: 21 Apr 2019, 09:43
00:00
A
B
C
D
E

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  25% (medium)

Question Stats:

74% (01:45) correct 26% (02:18) wrong based on 23 sessions

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if the least common multiple of positive integer z and 24 is 72, and GCD of z and 54 is 9,what is value of z?

A.3
B.6
C.9
D.18
E.36

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Originally posted by axezcole on 21 Apr 2019, 09:08.
Last edited by Bunuel on 21 Apr 2019, 09:43, edited 1 time in total.
Renamed the topic and edited the question.
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Re: if the least common multiple of positive integer z and 24 is 72, and G  [#permalink]

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New post 21 Apr 2019, 09:43
1
Given that,
LCM z and 24 = 72
GCD of z and 54 = 9

Condition 1: LCM z and 24 = 72
24 = \(2^{3} 3^{1}\)
72 = \(2^{3} 3^{2}\)
Therefore, z = \(3^{2}\), \(2^{1} 3^{2}\), \(2^{2} 3^{2}\), or \(2^{3} 3^{2}\), i.e. four possible values. For each of these 4 values of z and 24, the LCM is 72.

Condition 2: GCD of z and 54 = 9
54 = \(2^{1} 3^{3}\)
9 = \(3^{2}\)
Therefore, z = \(3^{2}\) x Muliple of any Prime number except 2.

From conditions 1 and 2, we get a unique value of z, i.e.z = \(3^{2}\)

Hence, the correct answer is option C. 9

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if the least common multiple of positive integer z and 24 is 72, and G  [#permalink]

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New post 21 Apr 2019, 18:14
1
The easiest way I found was to use the answers, we always start with C.

LCM of 9 and 24 is 72.

GDF of 9 and 54:

54: 1 2 3 6 9 18 27 54
9: 1, 3, 9

C
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Re: if the least common multiple of positive integer z and 24 is 72, and G  [#permalink]

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New post 21 Apr 2019, 19:01
2
1
axezcole92 wrote:
if the least common multiple of positive integer z and 24 is 72, and GCD of z and 54 is 9,what is value of z?

A.3
B.6
C.9
D.18
E.36



Just one line will eliminate all choices..

GCD of z and 54 is 9.
This tells you two things..
(i) z is a multiple of 9 but not of 27 as 54 contains a 27 in it......Only C, D and E are left
(ii) z is not even as 54 contains a 2 in it... Only A and C are left

Thus C is the answer
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Re: if the least common multiple of positive integer z and 24 is 72, and G   [#permalink] 21 Apr 2019, 19:01
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if the least common multiple of positive integer z and 24 is 72, and G

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