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# If the positive integer N is a perfect square, which of the following

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Intern
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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, 03:43
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, 01:39
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.

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.
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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, 11:41
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, 09:44
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, 18:52
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.

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?
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Joined: 02 Sep 2009
Posts: 52390
Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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04 May 2016, 21:56
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.

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.
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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, 01:05
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.

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?
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Joined: 02 Sep 2009
Posts: 52390
Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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05 May 2016, 01:11
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.

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.
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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, 20:26
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, 00:19
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.

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!
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Joined: 02 Sep 2009
Posts: 52390
Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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24 Apr 2017, 00:21
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.

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.
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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, 04:56
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, 06:07
sarathgopinath92 wrote:
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.

Factor is a positive divisor (at least on the GMAT).
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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, 08:32
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.

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 &nbs [#permalink] 03 May 2018, 08:32

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