Is the positive integer N a perfect square? (1) The number

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 help me. This is the question of 700 level from MGMAT CAT1

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.

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Please provide your reasons and explanations. Thank you.

I am having trouble understanding this reasoning. If we are interested in sum of factors, why are you talking about product of odd prime factors?
Thanks.

Take an example N = \(2^4 * 3^2 *5^4\)
If we write down its factors, we get: 1, 2, 3, 4, 5, 6, 8, 10, 12..... total 75 factors.
Lets just consider the odd factors i.e. factors made by \(3^2 *5^4\) (including 1)
These will be a total of 3*5 = 15 odd factors.
When you add an odd number of odd factors, result will be an odd number.
The rest of the factors of N will be even i.e. they will include at least one 2. their sum will definitely be even because no matter how many even numbers you add, you always get an even result.
Adding an odd number to an even number, you will get an odd number. Hence, sum of all factors of a perfect square will always be odd.

I do understand now.
However still not sure why perfect squares have "EVEN number of Even-factors".
I understand that power of the only even prime factor "2" determines the number of even factors. So, if a number is a perfect square then all the powers will be even. Then by adding 1 to all powers we get odd numbers, which we multiply. Then the number of even factors will be odd.

Tips about the perfect square:
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;

I think we should mention that the unwritten assumption is that we are only talking about positive factors. If we include -ve factors, the number of factors of every integer is always even.

Probably the best way of solving would be making the chart of perfect squares and its factors to check both statements, but below is the algebraic approach if needed.

Couple of things:
1. Note that if \(n\) is a perfect square powers of its prime factors must be even, for instance: \(36=2^2*3^2\), powers of prime factors of 2 and 3 are even.

2. 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.

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. For instance 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).

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.

Answer: D.

There are some tips about the perfect square:
• The number of distinct factors of a perfect square is ALWAYS ODD.
• The sum of distinct factors of a perfect square is ALWAYS ODD.
• A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
• Perfect square always has even number of powers of prime factors.

Hope it helps.

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.

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.

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.