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Data Sufficiency Error on MGMAT Cat Exam?

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Data Sufficiency Error on MGMAT Cat Exam?

by barrelbowl » Sun Nov 06, 2011 5:09 pm
Hey all,

Either I'm missing something here or there is an error in the following question. This was from CAT exam #4:

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.

The answer is D, both are sufficient.

I answered E, reason being -

If the integer was 3, then it would have an even number of factors (3 and 1). That goes for any prime number as well (i.e. 5 - 5 and 1). These numbers are not perfect squares, so (1) is insufficient.

For (2) using a prime number like 3 - 3+1 = 4 which is even. For a number the number 9 - the number of factors is also even (3+1 = 4). So N isn't necessarily a perfect square.

Are my definitions screwed up or is there something wrong with this question? Thanks in advance.
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by neelgandham » Sun Nov 06, 2011 5:49 pm
Is the positive integer N a perfect square?

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

If N can be written in the form (a^n)*(b^m)*(c^o), where a,b,c are distinct prime numbers and n,m,o are non-negative integers, then the total number of factors of N is (n+1)*(m+1)*(o+1) which is Even(From the choice). From the above statement we can deduce that at least one of the numbers among n,m,o is odd. If any one among the numbers n,m,o is odd, then the number N is not a square.

For e.g. Say N is equal to 1800 and can be written of the form (2^3)*(3^2)*(5^2).The total number of factors of N is equal to 4*3*3 which is an even number. So the number is not a perfect square.

Say N is equal to 3600 and can be written of the form (2^4)*(3^2)*(5^2).The total number of factors of N is equal to 5*3*3 which is an odd number. So the number is a perfect square.


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

All perfect squares have odd number of factors including 1 and have even number of factors excluding 1.

The Sum of Even number of Even numbers(2+4+6+8 = 20) is Even.
The Sum of Even number of Odd numbers (1+3+5+7 = 16) is also Even.

i.e. Irrespective of what type of a factor it is(Even or Odd), the sum of all the factors of a perfect square is an even number(Excluding 1). So, the sum of all factors of a perfect square is Odd (Including 1). From option 2, we can tell that the number N is not a square. Hence, Sufficient

Option D

p.s: Can you please post Data Sufficiency Questions here -> https://www.beatthegmat.com/data-sufficiency-f7.html
Last edited by neelgandham on Sun Nov 06, 2011 6:48 pm, edited 1 time in total.
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by GmatMathPro » Sun Nov 06, 2011 6:11 pm
There's no error.

1. Perfect squares always have an odd number of factors. Factors always come in pairs, so the number of factors is usually a multiple of 2, which makes it even. But with perfect squares, exactly one of the factors is paired with itself, so they have an odd number of factors. This statement allows you to answer the question with a definitive no.

2. The sum of all distinct factors of perfect squares is always odd. In your counterexample of 9, you forgot to add 9. The sum of the factors of 9 is 1+3+9=13.

Case 1: an odd perfect square has an odd number of factors that are all odd. the sum of an odd number of odds is always odd.

Case 2: An even perfect square will have an even number of even factors and an odd number of odd factors. If you examine the prime factorization of an even perfect square it will have the form 2^a*3^b*5^c.... where a, b, and c,... are all even. It will have a*(b+1)*(c+1)*... even factors. a is even, so a*(b+1)*(c+1)*... is even, so it has an even number of even factors. Therefore it must have an odd number of odd factors if the total number of factors is odd. An even number of even numbers added together is even. An odd number of odd numbers added together is odd. Even+odd=odd, so the sum of all factors of an even perfect square is always odd.

Therefore, statement 2 is sufficient to say no.
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by ArunangsuSahu » Sun Nov 06, 2011 6:32 pm
Distinct or PRIME factors?

If DISTINCT 'D' is the correct answer

1. Perfect squares have always odd number of distinct factors. 9 has 1,3,9. 25 ha 1,5,25. So SUFFICIENT. Not a Square
2. Sum of all the distinct factors= even. For perfect squares this sum is always odd. Add the factors in the Choice 1. So Sufficient. Not a Square
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by barrelbowl » Sun Nov 06, 2011 6:38 pm
Thanks guys - makes sense.

It is hard to believe the folks at MGMAT would have made a mistake.

And neelgandham sorry about that. Will def post any future DS questions in that forum - still familiarizing myself with this board!
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