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Which of the following must be true if the square root of X

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th03
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Which of the following must be true if the square root of X [#permalink] New post   21 Dec 2012, 03:44
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Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

Show Spoiler
Official answer is D.

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!
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Re: Which of the following must be true if the square root of X [#permalink] New post   21 Dec 2012, 04:06
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th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

Official answer is D.

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!


Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

Tips about perfect squares >0:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Answer: D.

Hope it helps.
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Re: Which of the following must be true if the square root of X [#permalink] New post   21 Dec 2012, 04:17
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

Show Spoiler
Official answer is D.

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!


I cannot be true since number of distinct factors of a square number is always odd. So we need to check only III. If III is true answer is D else answer can only be B.

Sum of distinct factors of a perfect square is always odd. Hence answer is D.

To answer your question, I believe the GMAT does not consider negative factors when it talks about factors of a number.
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Re: Which of the following must be true if the square root of X [#permalink] New post   22 Dec 2012, 01:28
Bunuel wrote:
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

Official answer is D.

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!


Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

Tips about perfect squares >0:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Answer: D.

Hope it helps.


Thanks Bunuel, Could you please clarify the term "Distinct Factors"?
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Re: Which of the following must be true if the square root of X [#permalink] New post   22 Dec 2012, 05:02
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Drik wrote:
Thanks Bunuel, Could you please clarify the term "Distinct Factors"?


Not sure what to clarify: distinct=different, so for example distinct factors of 8 are 1, 2, and 8.
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Re: Which of the following must be true if the square root of X [#permalink] New post   19 Jun 2013, 04:52
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: Which of the following must be true if the square root of X [#permalink] New post   23 Jun 2013, 22:41
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.


Hey Bunuel or others:

Could you please show a proof for this?

Thanks
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Re: Which of the following must be true if the square root of X [#permalink] New post   24 Jun 2013, 00:10
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mattce wrote:
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.


Hey Bunuel or others:

Could you please show a proof for this?

Thanks


Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.
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Re: Which of the following must be true if the square root of X [#permalink] New post   24 Jun 2013, 00:15
Bunuel wrote:
mattce wrote:
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.


Hey Bunuel or others:

Could you please show a proof for this?

Thanks


Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.


Haha, yeah I know that it's true by doing examples -- I was hoping for a formal proof though, if possible?
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Re: Which of the following must be true if the square root of X [#permalink] New post   24 Jun 2013, 00:19
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mattce wrote:
Bunuel wrote:
mattce wrote:

Hey Bunuel or others:

Could you please show a proof for this?

Thanks


Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.


Haha, yeah I know that it's true by doing examples -- I was hoping for a formal proof though, if possible?


You can do it yourself using the formula for the sum of the factors given here: math-number-theory-88376.html
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Re: Which of the following must be true if the square root of X [#permalink] New post   27 May 2015, 14:14
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Which of the following must be true if the square root of X is a positive integer?
This tells us that X is a perfect square. It will therefore not have an even number of distinct factors and we can eliminate A, C, and E.
The difference between B and D is choice III so we evaluate that. Since perfect squares always have an odd number of distinct factors, the sum of the distinct factors will be odd. That leaves only choice D.


I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III
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Re: Which of the following must be true if the square root of X [#permalink] New post   10 Oct 2023, 09:11
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Re: Which of the following must be true if the square root of X [#permalink]
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