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Updated on: 01 Mar 2018, 01:10
Originally posted by sekharm2389 on 01 Mar 2018, 00:53.
Last edited by Bunuel on 01 Mar 2018, 01:10, edited 1 time in total.
Last edited by Bunuel on 01 Mar 2018, 01:10, edited 1 time in total.
Renamed the topic and edited the question.
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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If P is a prime number and Q is a positive integer, how many factors does P^2*Q have?
(1) The lowest number that has both P^2 and Q^3 as its factors is 5400
(2) P and Q have only one common factor
(1) The lowest number that has both P^2 and Q^3 as its factors is 5400
(2) P and Q have only one common factor
01 Mar 2018, 01:25
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If P is a prime number and Q is a positive integer, how many factors does P^2*Q have?
(1) The lowest number that has both P^2 and Q^3 as its factors is 5400.
This means that 5,400 is the least common multiple of P^2 (square of a prime) and Q^3 (perfect cube), so both P^2 and Q^3 are factors of 5,400. Factorize: \(5400 = 2^3*3^3*5^2\). P^2, which is a square of a prime, must be 5^2 (so P must be 5) because in any other case (say if P^2 is 2^2 or 3^2), the remaining multiple will not be a perfect square (Q^3). Therefore, \(P^2 = 5^2\) (P = 5) and \(Q^3 = 2^3*3^3\) (Q = 6).
\(P^2*Q = 5^2*6 = 5^2*2*3\). The number of factors \(= (2 + 1)(1 + 1)(1 + 1) = 12\).
Sufficient.
(2) P and Q have only one common factor. This means that P and Q are co-prime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ...
Answer: A.
Hope it's clear.
(1) The lowest number that has both P^2 and Q^3 as its factors is 5400.
This means that 5,400 is the least common multiple of P^2 (square of a prime) and Q^3 (perfect cube), so both P^2 and Q^3 are factors of 5,400. Factorize: \(5400 = 2^3*3^3*5^2\). P^2, which is a square of a prime, must be 5^2 (so P must be 5) because in any other case (say if P^2 is 2^2 or 3^2), the remaining multiple will not be a perfect square (Q^3). Therefore, \(P^2 = 5^2\) (P = 5) and \(Q^3 = 2^3*3^3\) (Q = 6).
\(P^2*Q = 5^2*6 = 5^2*2*3\). The number of factors \(= (2 + 1)(1 + 1)(1 + 1) = 12\).
Sufficient.
(2) P and Q have only one common factor. This means that P and Q are co-prime, their only common factor is 1. This is clearly insufficient: for example, P could be 2 and Q could be 3, 3^2, 3^3, ...
Answer: A.
Hope it's clear.
srinjoy1990
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09 Mar 2018, 09:15
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Hi Bunuel,
Why are we assuming P^2 and Q^3 are the only factors apart from 1 for the number 5400?
I considered multiple possibilities for Q [2,3,6,1] so went with E. I think I am missing something in the reasoning. Can you help?
Tx.
Why are we assuming P^2 and Q^3 are the only factors apart from 1 for the number 5400?
I considered multiple possibilities for Q [2,3,6,1] so went with E. I think I am missing something in the reasoning. Can you help?
Tx.
