Is the positive integer N a perfect square?

[atomy1985]

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.

Please explain the how statement 2 is sufficient.. the explanation in the test review is a bit confusing.. i selected the correct answer in the test but i used hit and trial and used numbers to get the answers..please elaborate some more on what is mentioned in the MGMAT CAT explanation..

[goelmohit2002]

Can someone please tell why are we ignoring the negative factors....why the answer should not be E ?

IMO all the numbers have both positive and negative factors...

e.g. 4 has following factors: -4, -2, -1, 1, 2, 4

So all the numbers will have even number of factors(except zero) and sum of factors is always zero.....i.e. even...

Please tell what I am missing here ?

[DennaMueller]

Here is my understanding as to why the sum of the distinct factors of a perfect square will always be odd.

Start with how many distinct factors are in perfect square. There will always be an odd number of distinct factors for a perfect square, because the factors will be 1, the number itself and the 2 numbers that make it a perfect square. The 2 numbers that make it a perfect square are the same number, so they count as 1 distinct factor.

For Example:
49; 1, 7, 49
16; 1, 2, 4, 8, 16
9; 1, 3, 9

When you factor a perfect square you always have two of every prime factor. So, if there is an odd prime factor, there will always be two of them. The unique factors will include that odd prime and that odd prime times itself (which is odd). When you add these two distinct factors, you get an even. So, if you don't include 1, the sum of distinct factors will always be even. Obviously with the 1, the sum is odd.

Try it with the number 100.
100
10-10
2-5-2-5
1

The distinct factors are 2, 5, 10, 20, 50, 25,100 and 1
You have 2 odd - 5 & 25 plus you have 1.

[Ben Ku]

goelmohit2002, while technically negative integers can be factors, when we refer to the factors of a number, we mean the positive ones. So the factors of 6 are 1, 2, 3, 6.

Denna, I like the innovative solution to the question you present.

In a perfect square, there are even numbers of each prime factor. For example:
225 = 3^2 x 5^2
144 = 2^4 x 3^2

The distinct factors of 144 are all the combinations of the products of these prime factors (we'll leave out 1 for now). To get ODD factors, we need to multiply odd numbers together.

For 225, we have 3, 5, 3x3, 5x5, 3x5, 3x3x5, 3x5x5, 3x3x5x5 --> 6 different odd factors.
For 144, we have 3 and 3x3 --> 2 different odd factors.

You'll see that for every perfect square that has an odd prime factor, there will be an even number of distinct odd factors. Therefore, the sum of all distinct factors (not including 1) of a perfect square will be even; including 1, then sum will be odd.

[aleman]

I understand why the sum of distinct factors of a perfect square is odd, but what about the number 8? It seems to me that it breaks the rule, since its factors are:

1, 2, 4, 8

The sum of all these is odd, yet 8 is not a perfect square.

What am I missing?

[isha.myname]

I think the question here is that "is the condition #2 sufficient to determine whether a number is perfect square or not" and because a perfect square (and not just only perfect squares) always has sum of distinct factors as odd..#2 is sufficient.

[esledge]

I understand why the sum of distinct factors of a perfect square is odd, but what about the number 8? It seems to me that it breaks the rule, since its factors are:

1, 2, 4, 8

The sum of all these is odd, yet 8 is not a perfect square.

What am I missing?

This is really a restatement of what isha.myname said, but to be more complete:

If the sum of all distinct factors of N is odd, N could be a perfect square (N = 9, 16, 49, 100, as Denna discussed above) or N could NOT be a perfect square (N = 8, as you point out)

If the sum of all distinct factors of N is even, N CANNOT be a perfect square.

Thus, (2) is sufficient because it rules out the possibility that N is a perfect square. It is sufficient because it returns a definite "No" answer.

[victorgsiu]

What about the perfect square 4?

Four makes (1) insufficient

Factors of 4:
1,2,4
# of distinct factors = 2

Confused why we are limiting ourselves to just perfect squares with an odd number of factors?

[Ben Ku]

What about the perfect square 4?

Four makes (1) insufficient

Factors of 4:
1,2,4
# of distinct factors = 2

Confused why we are limiting ourselves to just perfect squares with an odd number of factors?

I'm unsure about your question. The factors of 4 are 1, 2, and 4. There are three distinct factors.

Whenever you have an odd number of distinct factors, you have a perfect square. The reason is usually a number can be expressed as PAIRS of factors. In the case of a perfect square, one of pairs is itself. Therefore, perfect squares always have odd number of factors; all other integers have even number of factors.

Hope that makes sense.

[prapruet.w]

I think I got it now
Please correct me if i am misunderstood
Every Perfect square For example 4, 9 , 25

4 --> 1,2,4
9--> 1,3,9
25--> 1,5,25

all of them has an odd number of distinct factors right
So (1) is sufficient to answer that N is not a perfect square

Also
1+2+4 = 7
1+3+9 = 13
1+5+25 = 31

All of them sums up to odd number
So (2) is sufficient to answer that N is not a perfect square

What do u think?

[mschwrtz]

you got it.

[goenkavivek]

Dears,

what about 36?
it is a perfect square & it's distinct factors are 1, 2,3,4,6,12,18 & 36

the sum of 1+2+3+4+6+12+18+36 is even
This proves that sum of a perfect square can be odd or even, the answer choice should be (A)

No?

[chris121981]

goenvivek...u forgot 9

[jnelson0612]

Thanks Chris!

[awahid14]

I understand why the sum of distinct factors of a perfect square is odd, but what about the number 8? It seems to me that it breaks the rule, since its factors are:

1, 2, 4, 8

The sum of all these is odd, yet 8 is not a perfect square.

What am I missing?

This is really a restatement of what isha.myname said, but to be more complete:

If the sum of all distinct factors of N is odd, N could be a perfect square (N = 9, 16, 49, 100, as Denna discussed above) or N could NOT be a perfect square (N = 8, as you point out)

If the sum of all distinct factors of N is even, N CANNOT be a perfect square.

Thus, (2) is sufficient because it rules out the possibility that N is a perfect square. It is sufficient because it returns a definite "No" answer.

Is this right? is it A or D?