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If the square root of p^2 is an integer, which of the follow
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Updated on: 20 Jan 2013, 20:41
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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
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Originally posted by monir6000 on 19 Jan 2013, 21:31.
Last edited by monir6000 on 20 Jan 2013, 20:41, edited 2 times in total.



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Re: If the square root of p^2 is an integer
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19 Jan 2013, 21:48
Answer should be D and not B. Every square has odd number of factors.



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Re: If the square root of p^2 is an integer
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19 Jan 2013, 23:40
Yes, D has to be the answer.
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Re: If the square root of p^2 is an integer, which of the follow
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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?
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Re: If the square root of p^2 is an integer, which of the follow
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08 Mar 2014, 06:29
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.
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Re: If the square root of p^2 is an integer, which of the follow
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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 Please let me kow what i am approaching wrong



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Re: If the square root of p^2 is an integer, which of the follow
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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. Hence D is the answer.
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Re: If the square root of p^2 is an integer, which of the follow
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02 Jun 2014, 01:04
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 1,2,3,4,6,9,18,36Please 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.
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Re: If the square root of p^2 is an integer, which of the follow
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02 Jun 2014, 01:06
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. Hence D is the answer. This would be correct if we were told that p is a positive integer greater than 1. Check here: ifthesquarerootofp2isanintegerwhichofthefollow146066.html#p1341427
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