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Director  Joined: 22 Mar 2011
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Re: Is the positive integer N a perfect square?  [#permalink]

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ashish8 wrote:
How is B sufficient? Sum of distinct factors of a perfect square is odd, but if n is 2, then the sum is also odd.

(2) states: The sum of all distinct factors of N is even.
Since the sum of distinct factors of a perfect square must be odd, we can conclude that N is not a perfect square.
So, the answer to the question "Is N a perfect square?" is a definite NO.
Therefore, (2) sufficient.

Not only perfect squares have the sum of their distinct factors odd. As you mentioned, for 2, the sum of its factors is odd, and it is not a perfect square.
So, if a number is a perfect square, then the sum of its factors is necessarily odd, but the reciprocal is not true. Meaning, if the sum of the factors is odd, the number is not necessarily a perfect square, it might be or not. But if the sum of the distinct factors is even, then certainly the number cannot be a perfect square.
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Re: Is the positive integer N a perfect square?  [#permalink]

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1
ashish8 wrote:
How is B sufficient? Sum of distinct factors of a perfect square is odd, but if n is 2, then the sum is also odd.

Also check this:

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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cipher wrote:
Pkit wrote:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

I have got this question wrong , but I would argue with the OA provided by MGMAT.

OA D
.

1) The number of distinct factors of N is even.

Suppose N = 4. It has 3 distinct factors: 1, 2 and 4.
Suppose N = 9. It has 3 distinct factors: 1, 3 and 9.
Suppose N = 16. It has 5 distinct factors: 1, 2, 4, 8, and 16.
Suppose N = 64. It has 7 distinct factors: 1, 2, 4, 8, 16, 32, and 64.
But that not the case. In fact, the case is opposite. So it is sufficient because N is not a square.

(2) The sum of all distinct factors of N is even.

If you follow the above pattern, you see 1 is always there. The sum of all distinct factors except 1 of N is even. If you add 1 on the even sum, that odd. So N is not a square.
But that not the case. In fact, the case is opposite. So it is sufficient because N is again not a square.

So D.

PS: A perfect square always have odd number of factors, for e.g a integer $$n$$ and its square $$n^2$$

Now, $$n$$ will have always have even number of factors, (take any number and you will realise that factors come in pairs), now $$n^2$$ will have all factors which $$n$$ has + one more which is $$n^2$$

Hi there,

I have a problem with this method. I think it is flawed but luckily works here.
We can see that the two statements should be true for perfect squares, but in no way have we proved that it is not true for non-perfect square.
For instance, getting a few examples of perfect squares and seeing that the sum of the factors is always odd, doesn't imply that summing the factors of a non-perfect square would not be odd...

The only way to properly answer is to know the properties given by Bunuel IMO.
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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Bunuel wrote:
tingle15 wrote:
I have a doubt...

Consider N=18, Its factors are: 1, 2, 3, 6, 9, 18. The sum of factors is 39 which is odd... Am i missing something?

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.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say $$n=a^p*b^q*c^r$$, given that the number of factors of $$n$$ is even --> $$(p+1)(q+1)(r+1)=even$$. But as we concluded if $$n$$ is a perfect square then powers of its primes $$p$$, $$q$$, and $$r$$ must be even, and in this case number of factors would be $$(p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd*odd*odd=odd\neq{even}$$. Hence $$n$$ can not be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if $$n$$ is a perfect square then (according to 3) sum of odd factors would be odd and sum of even factors would be even, so sum of all factors of perfect square would be $$odd+even=odd\neq{even}$$. Hence $$n$$ can not be a perfect square. Sufficient.

Hope it helps.

Hi, could you explain why " A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors" is true?
Thanks
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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Hi, could you explain why " A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors" is true?
Thanks

Here is a post that explains this: http://www.veritasprep.com/blog/2010/12 ... t-squares/
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square

Then how can D be the answer?
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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ssriva2 wrote:
Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square

Then how can D be the answer?

(2) says that the sum of all distinct factors of N is even.

If N = 4, then its factors are 1, 2, and 4 --> the sum = 1 + 2 + 4 = 7 = odd.

If N = 8, then its factors are 1, 2, 4 and 8 --> the sum = 1 + 2 + 4 +8 = 15 = even.
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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goldeneagle94 wrote:
Interesting Question !!!

A few facts to review:

A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Using the above facts, you can conclude that both statements are sufficient to answer the question.

Perfect Square 36:
(6 x 6)
(3 X 3 X 2 X 2)
4 total factors, 2 distinct factors, and sum is even…?
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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Hi Kitzrow,

You have to note the difference between "factors" and "prime factors"

36 has the following FACTORS:
1 and 36
2 and 18
3 and 12
4 and 9
6

So, there are 9 factors and the sum of those factors is 91.

This example matches the prior statements - 36 has an ODD number of factors and the sum of those factors is ODD.

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GMAT 1: 760 Q48 V46 Re: Is the positive integer N a perfect square?  [#permalink]

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Question here:

I thought perfect squares always have an even sum of powers of prime factors? For instance, in the example, 36 = 2^2 * 3^2, powers of 2 + 2 = 4. Yet when you look at the total number of factors (2+1) * (2+1) you get 9...and the explanation nulls statement 1 because there are an odd number of factors... Can anyone elaborate on this? Thank you!
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Re: Is the positive integer N a perfect square?  [#permalink]

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HunterJ wrote:
Question here:

I thought perfect squares always have an even sum of powers of prime factors? For instance, in the example, 36 = 2^2 * 3^2, powers of 2 + 2 = 4. Yet when you look at the total number of factors (2+1) * (2+1) you get 9...and the explanation nulls statement 1 because there are an odd number of factors... Can anyone elaborate on this? Thank you!

You are missing 1 important point.

When you look at number of factors of a perfect square you do ALL factors including 1 and the number itself.

Example, 25 = 5^2 ---> total number of factors = (2+1) =3 , (1,5,25). You can not just add the powers of perfect squares to get the total number of factors.

Statement 1 is sufficient as it gives a straight "no" to the question" is n a perfect square" as all perfect squares will have odd number of total factors.

Hope this helps.
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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mbaMission wrote:
Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

Nice explanation Bunuel- we could also arrive at those rules by testing square roots and knowing the distinct factor equation

Statement 1

What Bunuel is demonstrating is that this condition cannot allow a perfect square- the number of distinct factors of a number is found by taking the prime factorization of the number and then adding 1 to all the exponents and then multiplying the product being the number of distinct factors. Notice the numbers 9, 49, 100

3^2 = 9
3^(2+1) = 3 distinct factors: 9, 3 ,1

49= 7^2
7^(2+1) = 3 distinct factors :1,7,49

10^2= 100
5^2 2^2= 100
5^(2+1) 2^(2+1) = 9 distinct factors: 100, 50, 25, 20, 10 , 5, 4 ,2 ,1

So we can clearly see any number with an even number of distinct factors cannot be a perfect square- all perfect squares will have an odd number of distinct factors

suff

Statement 2

We can just try out a few small values such as 25, 9 , 49 - clearly the sum of all distinct factors of any perfect square will be odd

suff

D
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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goldeneagle94 wrote:
Interesting Question !!!

A few facts to review:

A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Using the above facts, you can conclude that both statements are sufficient to answer the question.

Hi,

Please explain this statement.....A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Thank you
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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Hi

Can we consider '1' as perfect square and evaluate both the statements. Because 1 is also a perfect square. In this case also option will be D?
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Re: Is the positive integer N a perfect square? (1) The number  [#permalink]

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A perfect square represents the square of an integer.

Questions on the properties of a perfect square are quite common on the GMAT. So, it’s important that you devote enough time to learn the unique characteristics of perfect squares, so that you will be able to deal with these questions. A good way to start would be to memorise the first 25 perfect squares and observe patterns.

Two important properties that we will need to use to solve this question are:

1) A perfect square always has an odd number of distinct factors. The vice-versa is true as well; any number that has an odd number of distinct factors has to be a perfect square.
For example, the number 4 has 3 factors viz., 1,2 and 4; similarly, the number 16 has 5 factors viz. 1, 2, 4, 8 and 16.

2) A perfect square always has an odd number of odd factors and an even number of even factors. This means that the sum of the factors of a perfect square is going to be odd.

Taking the same numbers, 4 and 16, we see that the sum of the factors of 4 is 7 and the sum of the factors of 16 is 31.

If there are odd number of odd factors, the sum of these will be odd; even number of even factors will add up to an even number. Adding an odd sum with an even sum will give us the final sum as odd. That is how the sum of the factors of a perfect square is always odd.

Let us analyse the statements now.

Statement I alone says that the number of distinct factors of N is even. This only means that N is NOT a perfect square. We can answer the question with a NO. Remember that a NO is as good an answer as a YES, in a Yes-No type of DS question.

Statement I alone is sufficient. Possible answer options are A or D. Answer options B, C and E can be eliminated.

Statement II alone says that the sum of all distinct factors is even. Again, this means that N is NOT a perfect square and hence we can answer the question with a NO.
Statement II alone is sufficient. Answer option A can be eliminated.
The correct answer option is D.

Hope that helps!
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