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If p is a positive integer is √p a positive integer?

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If p is a positive integer is √p a positive integer?  [#permalink]

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New post 19 Dec 2017, 22:16
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If p is a positive integer is √p a positive integer?

1) p is n percentage of n where n is a positive integer
2) p/100 is a positive integer
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Re: If p is a positive integer is √p a positive integer?  [#permalink]

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New post 19 Dec 2017, 22:49
(1) p is n% of n, this means p = n/100 * n = (n^2)/100
Here n is a positive integer, so n^2 is a perfect square. And n^2 must be divisible by 100, because then only p will be an integer. Now if we take square root both sides, we have
√p = √(n^2)/ √100 = n/10
Since n^2 is divisible by 100, this means n must be divisible by 10. So √p is an integer. Sufficient.

(2) p is divisible by 100.
If p = 400, then √p = 20 is an integer.
If p = 300, then √p = 10 √3, is not an integer. So Insufficient.

Hence A answer
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Re: If p is a positive integer is √p a positive integer?  [#permalink]

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New post 19 Dec 2017, 23:34
If p is a positive integer is √p a positive integer?

(1) p is n percentage of n where n is a positive integer --> p = n/100*n = (n/10)^2. Now, n is an integer, so n/10 is either an integer or a fraction but since p is an integer and fraction^2 cannot be an integer, then n/10 can only be an integer, thus p = (n/10)^2 = integer^2. Sufficient.

(2) p/100 is a positive integer. If p = 100, then the answer is YES but if p = 200, then the answer is NO. Not sufficient.

Answer: A.

Hope it's clear.
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If p is a positive integer is √p a positive integer?  [#permalink]

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New post 12 Sep 2018, 02:44
amanvermagmat wrote:
(1) p is n% of n, this means p = n/100 * n = (n^2)/100
Here n is a positive integer, so n^2 is a perfect square. And n^2 must be divisible by 100, because then only p will be an integer. Now if we take square root both sides, we have
√p = √(n^2)/ √100 = n/10
Since n^2 is divisible by 100, this means n must be divisible by 10. So √p is an integer. Sufficient.

(2) p is divisible by 100.
If p = 400, then √p = 20 is an integer.
If p = 300, then √p = 10 √3, is not an integer. So Insufficient.

Hence A answer


Why are we not considering -n/10 as a possible root of n^2/100, in which case p would be negative and give a NO to the "is p a positive integer" question?

That would make Statement 1 insufficient.
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If p is a positive integer is √p a positive integer?   [#permalink] 12 Sep 2018, 02:44
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If p is a positive integer is √p a positive integer?

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