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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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30 Jul 2014, 04:43
thanks for the explanation Bunuel completely missed that 49 is a factor of itself!!!



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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) / [ (a1)(b1)(c1) = (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.



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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.....



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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
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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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?



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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.
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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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?



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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.
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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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!



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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.
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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 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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If the positive integer N is a perfect square, which of the following [#permalink]
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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.



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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02 May 2017, 07:07



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Re: If the positive integer N is a perfect square, which of the following [#permalink]
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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 Oddfactors, and EVEN number of Evenfactors. 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
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