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If x is a prime number, what is the number of different factors of 18x

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roastedchips
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If x is a prime number, what is the number of different factors of 18x [#permalink] New post   06 Jul 2017, 07:29
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If x is a prime number, what is the number of different factors of 18x?
1) x^2 has 3 factors
2) x>3
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Re: If x is a prime number, how many different factors has 18x? [#permalink] New post   29 Jul 2018, 01:58
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KarthikvsGMAT wrote:
If x is a prime number, how many different factors has 18x?

1. \(x^2\) has 3 factors
2. \(x>3\)


OA: B
\(x\) is prime, how many different factors are of \(18x\)?
\(18x = 2*3^2*x\)
Case 1: if \(x =2\), then \(18x =36=2^2*3^2\)
then number of factors \((2+1)(2+1)=9\)
Case 2: if \(x =3\), then \(18x =54=2*3^3\)
then number of factors \((1+1)(3+1)=8\)
Case 3: if \(x >3\), then \(18x=2*3^2*x^1\)
then number of factors \((1+1)(2+1)(1+1)=12\)

Statement 1 :\(x^2\) has 3 factors
Square of every prime number has 3 factors: \(1,x,x^2\)
\(18 x\) can have \(9,8\) or \(12\) factors depending on whether \(x =2,3\) or \(x>3\)
So Statement 1 alone is not sufficient.

Statement 2: \(x>3\)
If \(x>3\), then \(18x\) will have \(12\) factors.
So Statement 2 alone is sufficient.
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Re: If x is a prime number, what is the number of different factors of 18x [#permalink] New post   06 Jul 2017, 09:01
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If x is a prime number, what is the number of different factors of 18x?

18x can be written as 2 * 3^2 * x

Case 1: If x=2, then 18x = 2^2 * 3^2 , total number of factors = 3*3 = 9
Case 2: If x=3, then 18x = 2 * 3^3 , total number of factors = 2*4 = 8
Case 3: If x= a prime number other than 2 and 3, then 18x = 2 * 3^2 * x , total number of factors = 2*3*2 = 12

So we need to find out what is the value of x?

Statement 1) x^2 has 3 factors

Square of any Prime number will have 3 factors. Eg. 2^2, 3^2, 5^2. As this statement gives no new information. This statement is not sufficient.

Statement 2) x>3

This statement clearly tells us that x is greater than 3, which means the third case is applicable here.So the total number of factors = 12. This statement is sufficient.

Answer is B
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If x is a prime number, what is the number of different factors of 18x [#permalink] New post   16 Jul 2018, 19:14
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roastedchips wrote:
If x is a prime number, what is the number of different factors of 18x?
1) x^2 has 3 factors
2) x>3


the answer would depend on if x is a factor of 18, that is if x is 2 or 3
If x is 2 or 3, Ans will be different ..
    a) x=2, 18*2=\(2^2*3^2....(2+1)(2+1)=3*3=9\)
    b) X=3, 18*3=2*3^3.......(1+2)(3+1)=3*4=12
If x is any other prime number, Ans will be different.
    \(18x=2*3^2*X.......(1+1)(2+1)(1+1)=2*3*2=12\)

1) \(x^2\) has 3 factors.
Just tells us that \(x^2 is square of a prime number\)
This is already known
Insufficient

2) X>3
now X is a prime number >3
Irrespective of the value of X, factors of 18x will be
18x=2*3^2*X
So factors are (1+1)(2+1)(1+1)=2*3*2=12
Sufficient

B
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Re: If x is a prime number, what is the number of different factors of 18x [#permalink] New post   31 Aug 2023, 19:23
Hi,
In these equations
Case 1: If x=2, then 18x = 2^2 * 3^2 , total number of factors = 3*3 = 9
Case 2: If x=3, then 18x = 2 * 3^3 , total number of factors = 2*4 = 8

How are you saying that total factor will be 9 or 8. What is this formula... I have never seen that.
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Re: If x is a prime number, what is the number of different factors of 18x [#permalink] New post   31 Aug 2023, 19:51
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shaishav.281281 wrote:
Hi,
In these equations
Case 1: If x=2, then 18x = 2^2 * 3^2 , total number of factors = 3*3 = 9
Case 2: If x=3, then 18x = 2 * 3^3 , total number of factors = 2*4 = 8

How are you saying that total factor will be 9 or 8. What is this formula... I have never seen that.



For finding number of positive factors, get the number in prime factorisation.
For example 72 = 2^3*3^2
Then, add one to each exponent and find their product => (3+1)(2+1) =12
Say number is \(x=a^p*b^q*c^r\), then number of factors are \((p+1)(q+1)(r+1)\)
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If x is a prime number, what is the number of different factors of 18x [#permalink] New post   02 Sep 2023, 05:22
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shaishav.281281 wrote:
Hi,
In these equations
Case 1: If x=2, then 18x = 2^2 * 3^2 , total number of factors = 3*3 = 9
Case 2: If x=3, then 18x = 2 * 3^3 , total number of factors = 2*4 = 8

How are you saying that total factor will be 9 or 8. What is this formula... I have never seen that.


Let's see if I can help.

There is a rule I learned on TTP to evaluate the total number of factors (a.k.a. number of divisors) of a number.
Let me give an example to demonstrate how the rule works. Suppose we have to find the number of factors of the number 2160. If we decompose this number into primes, we will have: 2^4 x 3^3 x 5^1. To find out the number of factors, we have to multiply the powers plus 1. In this case, the total number of factors will be (4+1)x(3+1)x(1+1) = 40 factors.

I hope that helps to comprehend the rule
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If x is a prime number, what is the number of different factors of 18x [#permalink]
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