|
Author |
Message |
|
TAGS:
|
|
|
Intern
Joined: 21 Oct 2012
Posts: 35
Location: United States
Concentration: Marketing, Operations
GMAT 1: 650 Q44 V35 GMAT 2: 600 Q47 V26 GMAT 3: 660 Q43 V38
GPA: 3.6
WE: Information Technology (Computer Software)
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
 30 Jul 2014, 04:43
thanks for the explanation Bunuel completely missed that 49 is a factor of itself!!!
|
|
|
|
|
|
Intern
Joined: 16 Mar 2014
Posts: 16
GMAT Date: 08-18-2015
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
24 Nov 2015, 02:39
Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. Let me try to explain it algebraically. The perfect number A is factored as : a^2 x b^2 x c^2 (the power is always EVEN, for the sake of simplicity, I use the power of 2).
1. The # of distinct factors = 3 x 3 x 3 = 27 : ODD 2. The sum of distinct factors = (a^3 - 1) (b^3 - 1) (c^3 - 1) / [ (a-1)(b-1)(c-1) = (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1) because a,b,c are different primes then there is at most one even factor among a, b, and c. Let's say a = 2 -> (a^2 + a + 1) = EVEN + EVEN + ODD = ODD b, c must be ODD -> (b^2 + b + 1) = ODD + ODD + ODD = ODD and (c^2 + c + 1) = ODD + ODD + ODD = ODD. SO (a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1) = ODD x ODD x ODD = ODD. 3. A = a^2 x b^2 x c^2 if a is EVEN and b, c are ODD. The # of possible factors of A = 3 x 3 x 3 = 27 = ODD. The # of possible factors not containing a (so will be ODD) = 3 x 3 = 9 = ODD. -> The # of possible factors containing a (so will be EVEN) is : 27 - 9 = 18 = ODD - ODD = EVEN. Hope it helps.
|
|
|
|
|
|
Intern
Joined: 27 Jul 2015
Posts: 1
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
29 Nov 2015, 12:41
Bunuel,
Confusion on why your second rule ("The sum of distinct factors of a perfect square is ALWAYS ODD") applies to the second statement. In the original question prompt, I do not see "distinct factors" being specified. I see "II. The sum of the factors of N is odd.", which is not always true (e.g. sum of factors of 9).
This may just be an error in original post, since I've seen a reply including the "distinct factor" description with statement 2 when repeating the question prompt, but just want to make sure I'm not missing anything.....
|
|
|
|
|
|
BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2642
GRE 1: 323 Q169 V154
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
16 Mar 2016, 10:44
here the rule used is => number of factors of any perfect square is odd =>the converse of the rule is also true. Hence C as the sum will be odd (1 is odd and rest sum will be even) also number of prime factors can be even or odd => discard statement 3 hence C
_________________
MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE! Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs!STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)AVERAGE GRE Scores At The Top Business Schools!
|
|
|
|
|
|
Intern
Joined: 26 Feb 2016
Posts: 12
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
04 May 2016, 19:52
Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. what about for (iii) if N = 1? then 1^2 = 1 therefore 0 even N = 2 then 2^2 = 4 1, 2, 4 2 even factors N = 3 then 3^2 = 9 1, 3, 9 0 even factors?
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 46146
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
04 May 2016, 22:56
sabxu1 wrote: Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. what about for (iii) if N = 1? then 1^2 = 1 therefore 0 even N = 2 then 2^2 = 4 1, 2, 4 2 even factors N = 3 then 3^2 = 9 1, 3, 9 0 even factors? 0 is an even integer.
_________________
New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!
Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.
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.
What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
|
|
|
|
|
|
Intern
Joined: 26 Feb 2016
Posts: 12
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
05 May 2016, 02:05
sabxu1 wrote: Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. what about for (iii) if N = 1? then 1^2 = 1 therefore 0 even N = 2 then 2^2 = 4 1, 2, 4 2 even factors N = 3 then 3^2 = 9 1, 3, 9 0 even factors? Yes thats what I mean...then isnt 3 true? why is only I and II true?
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 46146
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
05 May 2016, 02:11
sabxu1 wrote: sabxu1 wrote: Bunuel wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.
Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.
So I and II must be true. what about for (iii) if N = 1? then 1^2 = 1 therefore 0 even N = 2 then 2^2 = 4 1, 2, 4 2 even factors N = 3 then 3^2 = 9 1, 3, 9 0 even factors? Yes thats what I mean...then isnt 3 true? why is only I and II true? The question asks: which of the following must be true? III says: the number of distinct prime factors of N is even. III is not always true: a perfect square can have any number of prime factors. For example, 4 has only one prime factor, which is 2.
_________________
New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!
Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.
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.
What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
|
|
|
|
|
|
Director
Joined: 12 Nov 2016
Posts: 776
Location: United States
GRE 1: 315 Q157 V158
GPA: 2.66
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
19 Apr 2017, 21:26
Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the factors of N is odd.
III. The number of distinct prime factors of N is even.
A) I only
B) II only
C) I and II
D) I and III
E) I, II and III
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. If considered the perfect squares 4 and 9 4 has 3 distinct factors : 1, 2 (don't double count 2) , 4 The sum of these factors is 7 - and odd number 9 has 3 distinct factors : 1, 3 , 9 The sum of these factors is 13- an odd number Thus, I and II choice (C) is correct
|
|
|
|
|
|
Intern
Joined: 16 Jan 2017
Posts: 2
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
24 Apr 2017, 01:19
Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. Can't the factors be both positive and negative? For example the factors of 25 are -25,-5,-1,1,5,25 and the sum is 0,which is even. Correct me if I'm wrong!
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 46146
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
24 Apr 2017, 01:21
sarathgopinath92 wrote: Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. Can't the factors be both positive and negative? For example the factors of 25 are -25,-5,-1,1,5,25 and the sum is 0,which is even. Correct me if I'm wrong! No. A factor is a positive divisor.
_________________
New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!
Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.
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.
What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
|
|
|
|
|
|
Intern
Joined: 16 Jan 2017
Posts: 2
|
If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
02 May 2017, 05:56
Is that the case of GMAT? Well in general Mathematics, I'm sure factors can be either positive or negative. It's basically any number that divides a given number.
|
|
|
|
|
|
Math Expert
Joined: 02 Sep 2009
Posts: 46146
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
02 May 2017, 07:07
|
|
|
|
|
|
Intern
Joined: 08 Mar 2014
Posts: 6
|
Re: If the positive integer N is a perfect square, which of the following [#permalink]
Show Tags
03 May 2018, 09:32
Bunuel wrote: Orange08 wrote: If the positive integer N is a perfect square, which of the following must be true?
I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.
For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 only and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 only
Please write if my understanding is correct. Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors. Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors. So I and II must be true. Why are negative factors not considered in the total number of distinct factors? Eg - 25 # of + ve factors - 1,5 and 25 # of -ve factors - (-1), (-5) & (-25) Total number of factors = 6
|
|
|
|
|
|
|
Re: If the positive integer N is a perfect square, which of the following
[#permalink]
03 May 2018, 09:32
|
|
|
|
|
Go to page
Previous
1 2
[ 34 posts ]
|
|
|
|
|
|
close
