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If sqrt(x) is a positive integer, is sqrt(x) a prime number?

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If sqrt(x) is a positive integer, is sqrt(x) a prime number? [#permalink] New post 23 Mar 2006, 19:41
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If sqrt(x) is a positive integer, is sqrt(x) a prime number?

1: x is divisible by exactly three positive integers
2: all positive factors of x are odd
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 [#permalink] New post 23 Mar 2006, 21:46
1 st statement)
x can be 4 since it is divisible by 4,2,1
x can be 9 since it is divisible by 9,3,1
so insuff
2 st)
x can be 9 9,3,1
and 25 5 25 1
and 49 49 7 1
I am not sure but E
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 [#permalink] New post 23 Mar 2006, 23:12
Yurik79 wrote:
1 st statement)
x can be 4 since it is divisible by 4,2,1
x can be 9 since it is divisible by 9,3,1
so insuff
2 st)
x can be 9 9,3,1
and 25 5 25 1
and 49 49 7 1
I am not sure but E


You may not know it but if x=4 - sqrt(4) = 2 - prime number
same with x=9
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 [#permalink] New post 23 Mar 2006, 23:20
The solution is the following.

1. x is divisible by exactly three positive integers

any integer is divisable by itself and 1. This gives us 2 positive integers.

If there is only one third integer that divides x - it must be a prime number, because otherwise factors of that third integer would also divide x.

So x is divisable by 1, x and some prime.
That means x = (some prime)^n
But if n>2 x must be divisable by (some prime)^(n-1) - contradiction.
So far we found out, that x = (some prime)^2 => sqrt(x) = some prime

Sufficient

2. all positive factors of x are odd

pick x = 5*5*3*3 => sqrt(x) = 5*3 - not a prime number

Insufficient

The answer is A
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 [#permalink] New post 24 Mar 2006, 00:35
Should be A...

1) statement... X cannot be more than 10 since the figure can only have 3 multiples....

If x=9, sqrt X is a prime number..... x divisable by 1, 3 and 9

If x=4 sqrt X is also a prime number.... x divisable by 1, 2 and 4

If x=16 sqrt x is not a prime number but x has multiples 1, 2, 4, 8, 16 which cannot be the case according to the statement.

2) statement... insufficient. explanation post above :)
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 [#permalink] New post 24 Mar 2006, 00:54
1) x is divisible by exactly 3 positive integer --> not prime since prime numbers can only be divided by 1 and itself.

2) Not sufficient. Could be prime or not prime.

Ans A
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 [#permalink] New post 24 Mar 2006, 00:58
ywilfred wrote:
1) x is divisible by exactly 3 positive integer --> not prime since prime numbers can only be divided by 1 and itself.

2) Not sufficient. Could be prime or not prime.

Ans A


It is amazing how you interpreted the question and still got a correct answer :eyes2
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 [#permalink] New post 24 Mar 2006, 09:13
You are both pretty good in that department ;)

>1) statement... X cannot be more than 10 since the figure can only have 3 multiples....

Take 49 and tell me how many factros you got.
Or 12769.

SimaQ wrote:
ywilfred wrote:
1) x is divisible by exactly 3 positive integer --> not prime since prime numbers can only be divided by 1 and itself.

2) Not sufficient. Could be prime or not prime.

Ans A


It is amazing how you interpreted the question and still got a correct answer :eyes2
  [#permalink] 24 Mar 2006, 09:13
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