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If d is a positive integer, is d^(1/2) an integer?

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Bunuel
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If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 03:59
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If d is a positive integer, is \(\sqrt{d}\) an integer?

(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.


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If positive integer d equal to the square of an integer?

(1) d has an odd number of divisors.
(2) d has 3 divisors.
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D
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If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 04:10
If d is a positive integer, (d)^1/2 is an integer if d all prime factors of d have even power.

St. 1:
d = a^p * b^q .....
=> Number of factors of d = (p+1)(q+1)...

If number of factors = odd
=> (p+1)(q+1)... = odd
=> p, q ... are even

=> d = a^2n * b^2m...
=> d^1/2 = a^n * b^m...
=> d^1/2 is an integer.

=> St. 1 is sufficient.

St. 2:

d has three positive divisors
=> d = a^2
=> d^1/2 is an integer
=> St. 2 is sufficient.

Answer: D

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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 04:12
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If d is a positive integer, is an integer?

(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.

Statement 1:

As only perfect squares have odd no. of divisors, hence sufficient.

Statement 2:
3 positive divisors - again odd no. of divisors, perfect square.
Example: 4 - divisors (1,2,4)
9- divisors (1,3,9)
25 - divisors (1,5,25) and so on
Sufficient.

Hence (D)

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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 04:12
A perfect square had odd number of divisors.
For ex :- 4 has 3 divisors. 2^2 = (2+1) = 3
9 has 3 divisors, 3^2 = (2+1) = 3
And so on..
So both statements are sufficient alone..

IMO D

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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 04:18
I guess it is D

for 1: if d has 1 factor it is 1. root1 is an integer. if d has 3 divisors it is in the form of X^2. so x is an integer. if it has 5 divisors it is in the form of x^4, so X^2 (root of X^4) is an integer.

for 2: if d has 3 factors it is in the form of x^2. SO x is an integer.
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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 05:23
For Statement 1 & 2,

Only perfect squares have odd no. of divisors
Both are individually sufficient

IMO-D
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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   01 Jul 2020, 05:24
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Bunuel wrote:
If d is a positive integer, is \(\sqrt{d}\) an integer?

(1) d has an odd number of positive divisors.
(2) d has 3 positive divisors.


Question: is √d and Integer = Is d a Perfect square?

CONCEPT: Perfect Sqaures {1, 4, 9, 16, 25...} always have ODD number of factors


Statement 1: d has an odd number of positive divisors.

ie.. d is a perfect square

SUFFICIENT

Statement 2: d has 3 positive divisors.

i.e. d is square of a Prime number such as 4, 9, 25, 49... etc

SUFFICIENT

ANswer: Option D
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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink] New post   18 Oct 2023, 02:47
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Re: If d is a positive integer, is d^(1/2) an integer? [#permalink]
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