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Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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07 Mar 2014, 17:53

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Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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07 Mar 2014, 18:10

The first and third statements are clear. A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no. Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true? Is repetition of a prime factor counted in calculating even no. Of prime factors?

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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08 Mar 2014, 06:29

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AKG1593 wrote:

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors II. p^2 can be expressed as the product of an even number of prime factors III. p has an even number of factors

A. I B. II C. III D. I and II E. II and III

The first and third statements are clear. A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no. Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true? Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case: The square root of p^2 is an integer --> \(\sqrt{p^2}=integer\) --> \(p=integer\).

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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02 Jun 2014, 00:09

Bunuel wrote:

AKG1593 wrote:

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors II. p^2 can be expressed as the product of an even number of prime factors III. p has an even number of factors

A. I B. II C. III D. I and II E. II and III

The first and third statements are clear. A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no. Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true? Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case: The square root of p^2 is an integer --> \(\sqrt{p^2}=integer\) --> \(p=integer\).

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Hope it helps.

Hi Bunell,

Can you please tell whether while counting the total number of factors of a perfect square, do we count 1 and the number itself?

For example :- for any number lets say 6. the total number of factors will be 1,2,3,6 .

Isn't the case with perfect squares? as if we include 1 and number itself , the total number of factors of a perfect square will be even . like for 36- 1,2,3,4,6,9,18,36

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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02 Jun 2014, 00:52

monir6000 wrote:

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors II. p^2 can be expressed as the product of an even number of prime factors III. p has an even number of factors

A. I B. II C. III D. I and II E. II and III

I. p^2 is a perfect square and a perfect square has pairs of factors and '1' with it. The total number of factors is odd II. If p = 3 then p^2 = 9. The prime factors will always remain even as the p is an integer and the pairs have to come out of square root to make p an integer. III. This is not true always. Let us say p = 4, it has three factors: 1, 2 and 4. Hence it will not hold true always.

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

Show Tags

02 Jun 2014, 01:04

Expert's post

Manik12345 wrote:

Bunuel wrote:

AKG1593 wrote:

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors II. p^2 can be expressed as the product of an even number of prime factors III. p has an even number of factors

A. I B. II C. III D. I and II E. II and III

The first and third statements are clear. A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no. Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true? Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case: The square root of p^2 is an integer --> \(\sqrt{p^2}=integer\) --> \(p=integer\).

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Hope it helps.

Hi Bunell,

Can you please tell whether while counting the total number of factors of a perfect square, do we count 1 and the number itself?

For example :- for any number lets say 6. the total number of factors will be 1,2,3,6 .

Isn't the case with perfect squares? as if we include 1 and number itself , the total number of factors of a perfect square will be even . like for 36- 1,2,3,4,6,9,18,36

Please let me kow what i am approaching wrong

The total number of factors of any positive integer includes 1 and that integer itself. Why there should be an exception for perfect squares? Isn't a perfect square divisible by 1 and itself?

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36 --> 9 factors. _________________

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

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02 Jun 2014, 01:06

Expert's post

PerfectScores wrote:

monir6000 wrote:

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors II. p^2 can be expressed as the product of an even number of prime factors III. p has an even number of factors

A. I B. II C. III D. I and II E. II and III

I. p^2 is a perfect square and a perfect square has pairs of factors and '1' with it. The total number of factors is odd II. If p = 3 then p^2 = 9. The prime factors will always remain even as the p is an integer and the pairs have to come out of square root to make p an integer. III. This is not true always. Let us say p = 4, it has three factors: 1, 2 and 4. Hence it will not hold true always.

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