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If x is an integer, what is the sum of all distinct positive

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If x is an integer, what is the sum of all distinct positive [#permalink] New post 26 Nov 2010, 18:55
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If x is an integer, what is the sum of all distinct positive factors of \sqrt{x}?

(1) x has exactly 3 distinct positive factors
(2) x^2 -1 =3k where k is an odd integer

Please explain. I could solve this but it took me more than 3 mins.

Thanks
NAD
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Re: Sum of the distinct factors of square root of a number [#permalink] New post 26 Nov 2010, 20:08
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nades09 wrote:
If x is an integer, what is the sum of all distinct positive factors of \sqrt{x}?

(1) x has exactly 3 distinct positive factors
(2) x^2 -1 =3k where k is an odd integer

Please explain. I could solve this but it took me more than 3 mins.

Thanks
NAD


You can quickly solve it using logic. I will give you a teaser and see if you can arrive at the answer on your own.

Statement 1 tells you \sqrt{x} is prime.
Statement 2 tells you x^2 is even.
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Re: Sum of the distinct factors of square root of a number [#permalink] New post 26 Nov 2010, 20:32
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Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C
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Re: Sum of the distinct factors of square root of a number [#permalink] New post 27 Nov 2010, 11:16
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nades09 wrote:
Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C


You got most of it. If you go through factors theory, it will help you understand that only a square of a prime number can have 3 factors. e.g. 4 or 25 or 49 or 121 or 169..
1 , 2, 4 are factors of 4
1, 5, 25 are factors of 25
1, 7, 49 are factors of 49
1, 11, 121 are factors of 121 etc

So if x has 3 factors, \sqrt{x} must be prime.

Also from statement 2, x^2 = 3k + 1 where k is odd. So 3k is odd and 3k + 1 is even. So x^2 is even. Now, if x^2 is even, x has to be even too (It is not possible that a power of an odd number becomes even. If x is odd, x^2, x^3 etc all will be odd. If x is even, x^2, x^3etc all will be even.). Then \sqrt{x} must also be even.

The only number that is even and prime is 2. So \sqrt{x} must be 2.
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Re: If x is an integer, what is the sum of all distinct positive [#permalink] New post 12 Sep 2013, 08:26
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Re: If x is an integer, what is the sum of all distinct positive   [#permalink] 12 Sep 2013, 08:26
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