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The least common multiple of positive integers p and q is 72. What is

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The least common multiple of positive integers p and q is 72. What is [#permalink] New post   04 Feb 2020, 11:18
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The least common multiple of positive integers p and q is 72. What is the greatest common factor of p and q?

(1) pq is a perfect square
(2) pq is a perfect fourth power
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Re: The least common multiple of positive integers p and q is 72. What is [#permalink] New post   04 Feb 2020, 20:43
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sjuniv32 wrote:
The least common multiple of positive integers p and q is 72. What is the greatest common factor of p and q?

(1) pq is a perfect square
(2) pq is a perfect fourth power


PROPERTY:
LCM x HCF = Product of those two numbers

ie. p*q = 72*HCF

Question: GCD (HCF) = ? OR p*q = ?

Statement 1: p*q is a perfect square

i.e. p*q is a multiple of 72 and a perfect square as well

i.e. p*q = {144, 576, 1296... etc}

i.e. GCD = p*q/LCM = {144/72, 576/72, 1296/72... } = {2, 8, 18...}

Multiple possible values hence

NOT SUFFICIENT

Statement 2: pq is a perfect fourth power

i.e. p*q is a multiple of 72 and a perfect forth power

i.e. p*q = {\(1296, 1296*2^4\), ... etc.}

i.e. GCD = p*q/LCM = {\(1296/72\), \(1296*2^4/72\)... etc } = {18, 288.. etc.}

BUT GCD can NOT be greater than LCM hence only possible value of GCD = 18

SUFFICIENT

Answer: Option B
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The least common multiple of positive integers p and q is 72. What is [#permalink] New post   05 Feb 2024, 08:18
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How do you come up with numbers like 1296 on the spot? Did you choose 1296 as it's 6^4?
Can you explain the logic behind choosing 1296?
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The least common multiple of positive integers p and q is 72. What is [#permalink]
05 Feb 2024, 08:18
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