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

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25 Sep 2010, 10:37

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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 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

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.

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.

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

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14 Oct 2010, 05:09

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.

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.

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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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.

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.

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.

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.

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

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26 Sep 2012, 03:30

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.

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

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06 Oct 2012, 19:54

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.

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:

Tips about perfect squares: 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.

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

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07 Oct 2012, 05:15

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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07 Oct 2012, 08:57

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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Kudos is the currency of appreciation.

You can have anything you want if you want it badly enough. You can be anything you want to be and do anything you set out to accomplish, if you hold to that desire with the singleness of purpose. ~William Adams

Many of life's failures are people who did not realize how close to success they were when they gave up. ~Thomas A. Edison

Wir müssen wissen, Wir werden wissen. (We must know, we will know.) ~Hilbert

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

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30 Jul 2014, 03:35

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:

Tips about perfect squares: 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?

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:

Tips about perfect squares: 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.
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