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If the least common multiple of positive integer Z and 24 is 72, and

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axezcole
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If the least common multiple of positive integer Z and 24 is 72, and [#permalink] New post  Updated on: 28 Oct 2020, 10:14
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If the least common multiple of positive integer Z and 24 is 72, and the greatest common divisor of Z and 54 is 9, what is the 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 28 Oct 2020, 10:14, edited 2 times 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 [#permalink] New post  21 Apr 2019, 09:43
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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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jfranciscocuencag
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Re: If the least common multiple of positive integer Z and 24 is 72, and [#permalink] New post  21 Apr 2019, 18:14
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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 [#permalink] New post  21 Apr 2019, 19:01
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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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ScottTargetTestPrep
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Re: If the least common multiple of positive integer Z and 24 is 72, and [#permalink] New post  21 Mar 2020, 15:36
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axezcole 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


Since 3 and 6 are factors of 24, then the LCM of either of them and 24 can’t be 72. We can rule out A and B as the correct answer. Now let’s look at 9. Since 9 = 3^2 and 24 = 3 x 2^3, the LCM of 9 and 24 is 3^2 x 2^3 = 8 x 9 = 72. Since 9 is a factor of 54, the GCD of 9 and 54 is 9. So z must be 9.

Answer: C
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jacelynnp
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Re: If the least common multiple of positive integer Z and 24 is 72, and [#permalink] New post  21 Mar 2020, 21:27
i don't quite understand condition 1. how do you come to the conclusion that those 4 options are possible values of z?
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Re: If the least common multiple of positive integer Z and 24 is 72, and [#permalink]
21 Mar 2020, 21:27
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