Is the positive integer n equal to the square of an integer?

Question

(1) For every prime number p, if p is a divisor of n, then so is p^2
(2)  \sqrt{n}  is an integer

Bunuel · Accepted Answer

Is the positive integer n equal to the square of an integer?

Question: is  n=integer^2 ? So, basically we are asked whether  n  is a perfect square (a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square.).

(1) For every prime number p, if p is a divisor of n, then so is p^2
consider n=27 which is not a square, now if p=3 and p^2=9, both will divide 27, so this condition is not sufficient to determine if n is a square
(2) √n is an integer

If n is square then √n is always an integer, hence option B

(2)  \sqrt{n}  is an integer -->  \sqrt{n}=integer  -->  n=integer^2 . Sufficient.

Answer: B.

ashiima86 · Answer

ST 1-isnt this telling you all the prime factors of n are raised to even powers which makes n a square number-i got wrong can you please re-explain.

pritish2301 · Answer

Hi Ashima,

This is how I verified statement 1.

Take n = 36, p = 2
If p is a divisor of n, then so is p2. Check for p^2 => 2^2 => 4, 36 is divisible by 4. - OK

Take n = 15, p = 3
If p is a divisor of n, then so is p2. Check for p^2 => 3^2 => 9, 15 is not divisible by 9 - Not OK

Hence statement 1 is insufficient. Hope this is OK.

Maybe Bunuel can help us better.

Regards,
Pritish

Bunuel · Answer

No, the first statement says that if a prime number p is a factor of n, then so is p^2, which means that the power of p is more than or equal to 2: it could be 2, 3, ... So, n is not necessarily a prefect square. For example, if  n=2^2  then the answer is YES but if  n=2^3  then the answer is NO (notice that in both case prime number 2 as well as 2^2 are divisors of n, so our condition is satisfied).

Hope it's clear.

Bluelagoon · Answer

Is the positive integer n equal to the square of an integer?

(1) For every prime number p, if p is a divisor of n, then so is p^2
(2) root n is an integer

From 1 ) If p=4, than 16 is also a factor. Which can qualify n to be a perfect square.But if p=2 than 4 is also a factor. However we can't say if n is square of an integer or not. Hence Insufficient.

2) If root n is an integer -> N has to be the square of an integer. Sufficient.

Answer B.

PSD927 · Answer

(1) For every prime number p, if p is a divisor of n, then so is p^2
consider n=27 which is not a square, now if p=3 and p^2=9, both will divide 27, so this condition is not sufficient to determine if n is a square
(2) √n is an integer

If n is square then √n is always an integer, hence option B

Is the positive integer n equal to the square of an integer?

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