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

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Which of the following must be true if the square root of X [#permalink] New post 21 Dec 2012, 02:44
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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

[Reveal] 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!
[Reveal] Spoiler: OA

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Re: Which of the following must be true if the square root of X [#permalink] New post 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.

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, 03:17
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

[Reveal] 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, 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"?
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Re: Which of the following must be true if the square root of X [#permalink] New post 22 Dec 2012, 04:02
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Re: Which of the following must be true if the square root of X [#permalink] New post 19 Jun 2013, 03:52
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Re: Which of the following must be true if the square root of X [#permalink] New post 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.


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 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.

Hope it helps.
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COLLECTION OF QUESTIONS:
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Re: Which of the following must be true if the square root of X [#permalink] New post 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?
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Re: Which of the following must be true if the square root of X [#permalink] New post 23 Jun 2013, 23:19
Expert's post
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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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Which of the following must be true if the square root of X [#permalink] New post 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.

Will be grateful if you could clarify a little.

Thanks
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Re: Which of the following must be true if the square root of X [#permalink] New post 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).
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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