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If P is a prime number and Q is a positive integer, how many factors
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01 Mar 2018, 00:25
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1
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, ...
Re: If P is a prime number and Q is a positive integer, how many factors
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09 Mar 2018, 09:01
srinjoy1990 wrote:
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.
How can P^2 and Q^3 be the only factors of 5,400? A number to have four factors should be of the form prime1*prime2, in this case its factors would be 1, p1, p2, and p1*p2.
If you say that Q can 1, 2, or 3, then what would P and would 5,400 be the LCM of P^2 an Q^3?
_________________
Re: If P is a prime number and Q is a positive integer, how many factors
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09 Mar 2018, 10:03
Quote:
(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, ...
Are 2 prime numbers coprime? 2 and 7, for example. Thanks
_________________
If P is a prime number and Q is a positive integer, how many factors
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09 Mar 2018, 10:06
Akela wrote:
Quote:
(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, ...
Are 2 prime numbers coprime? 2 and 7, for example. Thanks
Re: If P is a prime number and Q is a positive integer, how many factors
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09 Mar 2018, 10:07
Akela wrote:
Quote:
(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, ...
Are 2 prime numbers coprime? 2 and 7, for example. Thanks
Two numbers are co-prime if they do not share any common factor but 1. So, any two different primes are co-prime. For example, 2 and 7 do not share any common factor but 1.
_________________
Re: If P is a prime number and Q is a positive integer, how many factors
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09 Mar 2018, 10:52
Bunuel wrote:
srinjoy1990 wrote:
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.
How can P^2 and Q^3 be the only factors of 5,400? A number to have four factors should be of the form prime1*prime2, in this case its factors would be 1, p1, p2, and p1*p2.
If you say that Q can 1, 2, or 3, then what would P and would 5,400 be the LCM of P^2 an Q^3?
Thanks got my error. The only possible combination is 6^3*5^2 because, the numbers have to be coprime [The composite number cannot contain the prime number here], so the LCM is effectively product of the two. Should have noticed. _________________
If P is a prime number and Q is a positive integer, how many factors
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10 Mar 2018, 10:33
Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything.
Re: If P is a prime number and Q is a positive integer, how many factors
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10 Mar 2018, 10:48
shu123 wrote:
Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything.
First of all, please re-read this: (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, ...
Next, try to pick P and Q so that they are co-prime but so that both are not primes.
Re: If P is a prime number and Q is a positive integer, how many factors
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13 May 2018, 01:00
1
If P is a prime number and Q is a positive integer, how many factors does P^2*Q have?
Statement 1:- (1) The lowest number that has both P^2 and Q^3 as its factors is 5400.
This means that 5,400 is the LCM 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) 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.
Statement 2 :- P and Q have only one common factor. This means that P and Q are co-prime,. For example, P could be 2. Q could be 3, 3^2, 3^3, ... There is no data provided for Q. This is clearly insufficient.
Re: If P is a prime number and Q is a positive integer, how many factors
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29 May 2018, 06:22
Bunuel wrote:
shu123 wrote:
Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything.
First of all, please re-read this: (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, ...
Next, try to pick P and Q so that they are co-prime but so that both are not primes.
Re: If P is a prime number and Q is a positive integer, how many factors
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29 May 2018, 06:30
@s wrote:
Bunuel wrote:
shu123 wrote:
Hi Bunuel, Regarding 2nd statement ,since P and Q are coprime,I guess the number of factors will not change because if we take P=2 and Q=3 then number of factors for P^2Q will be (2+1)(1+1)=6 and if we take P=2 and Q=7 then also number of factors will remain same.Kindly let me know if i am missing anything.
First of all, please re-read this: (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, ...
Next, try to pick P and Q so that they are co-prime but so that both are not primes.
This should help.
Hi Bunuel
Are 5 and 12 are co-primes ?
Thanks
Yes.
The factors of 5 are 1 and 5. The factors of 12 are 1, 2, 3, 4, 6 and 12.
As you can see the only factor they share is 1, so 5 and 12 are co-prime.
_________________
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45% (02:27) correct 
