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