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What is the total number of positive factors of a?

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What is the total number of positive factors of a?  [#permalink]

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New post Updated on: 25 Dec 2018, 08:40
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A
B
C
D
E

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Question Stats:

33% (02:21) correct 67% (01:46) wrong based on 38 sessions

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What is the total number of positive factors of a?

(1) 2a is a multiple of 50.

(2) a/9 is an integer and has exactly 3 positive factors.

Originally posted by kiran1213 on 24 Dec 2018, 11:02.
Last edited by chetan2u on 25 Dec 2018, 08:40, edited 1 time in total.
Corrected the OA
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Re: What is the total number of positive factors of a?  [#permalink]

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New post 25 Dec 2018, 08:39
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What is the total number of positive factors of a?

(1) 2a is a multiple of 50.
We do not know what is 2a..
It can be 50, that is 2a=50...a=25=5^2....prime factors is (2+1)=3
Say 2a is 100, that is 2a=100....\(a=50=2*5^2\). Factors is (1+1)(2+1)=2*3=6
Insufficient.

(2) a/9 is an integer and has exactly 3 positive factors.
This means a/9 is square of a prime number..
Let \(\frac{a}{9}=9.....a=81=3^4\).. factors =(4+1)=5
Let \(\frac{a}{9}=4.....a=36=3^2*2^2\).. factors =(2+1)(2+1)=3*3=9
Different answers, so insufficient..

Combined..
a is a multiple of 25.. but \(\frac{√a}{3}\) is a prime number..
So only possible value is 25*9..
Factors of \(5^23^2\) are (2+1)(2+1)=9..
C

OA given is wrong..
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Re: What is the total number of positive factors of a?  [#permalink]

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New post 25 Dec 2018, 04:52
kiran1213 wrote:
What is the total number of positive factors of a?

(1) 2a is a multiple of 50.

(2) a/9 is an integer and has exactly 3 positive factors.


Statement (1): 2a is a multiple of 50
Therefore a is a multiple of 50/2 = 25.
if a is a multiple of 25, a can be 25, 50, 75....so on.
Each of these numbers will have different numbers of positive factors.
Hence insufficient.

Statement (2): a/9 is an integer and has exactly 3 positive factors.
We can write this as a=9k, where k is an integer.
Or a=3^2 x k^m
It says that 'a' has exactly 3 positive factors. Therefore a=3^2 x 1 = 3^2 (because if 'k' is anything >1, then a will not have exactly 3 factors, but will have more than 3 factors)
Sufficient.

Hence, B.

I don't understand why OA is D.
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Re: What is the total number of positive factors of a?  [#permalink]

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New post 25 Dec 2018, 08:41
Darshi04 wrote:
kiran1213 wrote:
What is the total number of positive factors of a?

(1) 2a is a multiple of 50.

(2) a/9 is an integer and has exactly 3 positive factors.


Statement (1): 2a is a multiple of 50
Therefore a is a multiple of 50/2 = 25.
if a is a multiple of 25, a can be 25, 50, 75....so on.
Each of these numbers will have different numbers of positive factors.
Hence insufficient.

Statement (2): a/9 is an integer and has exactly 3 positive factors.
We can write this as a=9k, where k is an integer.
Or a=3^2 x k^m
It says that 'a' has exactly 3 positive factors. Therefore a=3^2 x 1 = 3^2 (because if 'k' is anything >1, then a will not have exactly 3 factors, but will have more than 3 factors)
Sufficient.

Hence, B.

I don't understand why OA is D.


Statement II will not be sufficient..
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Re: What is the total number of positive factors of a?  [#permalink]

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New post 25 Dec 2018, 09:02
chetan2u wrote:
What is the total number of positive factors of a?

(1) 2a is a multiple of 50.
We do not know what is 2a..
It can be 50, that is 2a=50...a=25=5^2....prime factors is (2+1)=3
Say 2a is 100, that is 2a=100....\(a=50=2*5^2\). Factors is (1+1)(2+1)=2*3=6
Insufficient.

(2) a/9 is an integer and has exactly 3 positive factors.
This means a/9 is square of a prime number..
Let \(\frac{a}{9}=9.....a=81=3^4\).. factors =(4+1)=5
Let \(\frac{a}{9}=4.....a=36=3^2*2^2\).. factors =(2+1)(2+1)=3*3=9
Different answers, so insufficient..

Combined..
a is a multiple of 25.. but \(\frac{√a}{3}\) is a prime number..
So only possible value is 25*9..
Factors of \(5^23^2\) are (2+1)(2+1)=9..
C

OA given is wrong..


This was a much needed clarification. Thank you chetan2u
I've also understood why B won't be sufficient. Many thanks again!
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Re: What is the total number of positive factors of a?   [#permalink] 25 Dec 2018, 09:02
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