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Manager  Joined: 25 Jul 2010
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If the positive integer N is a perfect square, which of the following  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 53% (01:37) correct 47% (01:42) wrong based on 606 sessions

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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.

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.

Originally posted by Orange08 on 25 Sep 2010, 10:37.
Last edited by Bunuel on 29 Apr 2020, 23:11, edited 3 times in total.
Reformatted
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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8
31
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 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

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

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2
1 is neither considered a prime nor a composite number

The answer to the question should be (1) & (2) only

A perfect square has an odd number of factors
The number of odd factors that a perfect square has is also odd
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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Thanks for the tips Bunuel. I had never thought about 2 and 3, but they make sense.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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1
On a similar note..
I. The number of factors can be found using the formula = (p+1)(q+1)(s+1).... where p,q,r are the indices

eg1: 25= 5^2 so the number of factors are (2+1) = 3
eg2: 24 = (2^3) * (3^1) so the number of factors are (3+1)(1+1) = 8
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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g4gmat wrote:
On a similar note..
I. The number of factors can be found using the formula = (p+1)(q+1)(s+1).... where p,q,r are the indices

eg1: 25= 5^2 so the number of factors are (2+1) = 3
eg2: 24 = (2^3) * (3^1) so the number of factors are (3+1)(1+1) = 8

To clarify, this requires the prime factorization of a number.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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Nice tips Thanks Bunuel
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Thank you! Bunuel & g4gmat for sharing your tips.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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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 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

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.

Hi Bunuel!, is there a way of picking numbers to solve this question?
I think that it would be better than trying to remember the rules about perfect squares during the test Thanks!
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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metallicafan 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 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

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.

Hi Bunuel!, is there a way of picking numbers to solve this question?
I think that it would be better than trying to remember the rules about perfect squares during the test Thanks!

Those are useful properties which are worth to remember, even better if you understand why they are right.

Questions about these properties with explanation why they are right:
help-factors-problem-99145.html?hilit=perfect%20square
perfect-square-94700.html?hilit=perfect%20square

As for picking numbers: you can easily prove that III is not always true as soon as you pick appropriate perfect square, say n=2^2=4 --> 4 has 1 (so odd) prime factor, which is 2. For I and II if you try 2-3 perfect squares you'll see that all of them will have the odd number of distinct factors and the odd sum of the distinct factors and though 2-3 examples do not prove that these statement are ALWAYS true you can make educated guess.

The question asks which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

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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Thank you Bunuel !

I was not aware of 3 and 4. Manager  Joined: 09 Jun 2011
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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These two facts about perfect square is applicable in this problem, so the Option I and II is true

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

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Bunuel,

Can you explain pt4 in detail. What if the question had III instead as "prime factors of N are always odd". I think the number prime factor for perfect square will always be odd.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.

That's correct:

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.

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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Is 0 considered a perfect square?
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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closed271 wrote:
Is 0 considered a perfect square?

A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

Since 0=0^2 then 0 is a perfect square. But the properties discussed do not apply to 0.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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Bunuel wrote:
closed271 wrote:
Is 0 considered a perfect square?

A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

Since 0=0^2 then 0 is a perfect square. But the properties discussed do not apply to 0.

That is what I wanted to point out - that those properties do not apply to 0. Thank you for confirming. However, the question being discussed mentions 'a positive integer', so it should be fine.
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Re: If the positive integer N is a perfect square, which of the following  [#permalink]

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Bunuel wrote:
doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.

That's correct:

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.

Hope it helps.

Hi Bunuel,

In statement 2 you say that the sum of distinct factors of a perfect square is ALWAYS odd but if we consider the perfect square 49 its factors are =7*7*1 and here the distinct factoros of 49 are 7 and 1 which sum to 8 an EVEN number. Can you please help me understand how statement 2 is always true?

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

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havoc7860 wrote:
Bunuel wrote:
doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.

That's correct:

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.

Hope it helps.

Hi Bunuel,

In statement 2 you say that the sum of distinct factors of a perfect square is ALWAYS odd but if we consider the perfect square 49 its factors are =7*7*1 and here the distinct factoros of 49 are 7 and 1 which sum to 8 an EVEN number. Can you please help me understand how statement 2 is always true?

Thanks,
Aamir.

Factors of 49 are 1, 7, and 49: 1 + 7 + 49 = 57 = odd.
_________________ Re: If the positive integer N is a perfect square, which of the following   [#permalink] 30 Jul 2014, 03:38

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