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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!

Re: Which of the following must be true if the square root of X [#permalink]
21 Dec 2012, 03:06

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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.

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. _________________

Did you find this post helpful?... Please let me know through the Kudos button.

Re: Which of the following must be true if the square root of X [#permalink]
22 Dec 2012, 00:28

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"?

Re: Which of the following must be true if the square root of X [#permalink]
23 Jun 2013, 21:41

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.

Re: Which of the following must be true if the square root of X [#permalink]
23 Jun 2013, 23:10

Expert's post

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.

Re: Which of the following must be true if the square root of X [#permalink]
23 Jun 2013, 23:15

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? _________________

Re: Which of the following must be true if the square root of X [#permalink]
11 Oct 2013, 12:12

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.

Hi Bunuel,

My doubt was regarding the distinct factors.

For example

we take 16 -

Wont we consider the negative factors also?

Like for eg - for 16 they would be -1, 1, -2, 2, -4, 4, -8, 8, -16, 16 so that gives us an even number of distinct factors right?

Why wont we consider the negative in this case. The integers/factors with the negative sign are distinct too.

Re: Which of the following must be true if the square root of X [#permalink]
12 Oct 2013, 08:36

Expert's post

abhishekgulshan wrote:

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.

Hi Bunuel,

My doubt was regarding the distinct factors.

For example

we take 16 -

Wont we consider the negative factors also?

Like for eg - for 16 they would be -1, 1, -2, 2, -4, 4, -8, 8, -16, 16 so that gives us an even number of distinct factors right?

Why wont we consider the negative in this case. The integers/factors with the negative sign are distinct too.

Will be grateful if you could clarify a little.

Thanks

Please read the red part in the post you quote: factor is a "positive divisor" (at least on the GMAT). _________________