Is it always the case that for an integer that is not a perfect square that that the sum of its factors is even? When I test 8, the sum of its factors is 15, which is odd. But when I have been testing other numbers the sum is always is even. Thanks.

Hi rjcesaro,
Definitely not. For example, 2 is not a perfect square, but the sum of its factors is 3 (1 + 2). Unfortunately, I don't think you can make this generalization.

Thank you,

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Jamie Nelson
ManhattanGMAT Instructor

Hi,

It is, however, the case when counting the total number of factors of a number. A prime number always has an odd number of total factors and a non-prime number always has an even number of total factors.

Absolutely not. Every prime has two factors, which is definitely not an odd number. The number of factors is only odd when the number is a perfect square..

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Tim Sanders
Manhattan GMAT Instructor

I have to disagree with this also.

A prime number always has an even number of factors: 1 and itself, or two factors.

A non-prime number may have an even or odd number of factors. For example, 6 has factors 1, 2, 3, and 6, or an even number of factors. 4 has factors 1, 2, and 4, or an odd number of factors. A perfect square will have an odd number of factors because the square root is counted only once.

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Jamie Nelson
ManhattanGMAT Instructor

oops! sorry i typed "prime" when what i meant was "perfect square" and "non prime" when i meant "not a perfect square" For example 9 is a perfect square, having factors 1,3,9 (total number factors= 3= an odd number) whereas 8, a non perfect square number, has factors 1,2,4,8 (total number of factors= 4= even number).

There is a simple way of calculating the total number of factors of a given number. We'll pick number 144, which can be expressed as a product of all its prime numbers. 144= 2*2*2*2*3*3, which can in turn be written as 2^4 *3^2. Now, the total number of factors is given by adding 1 to each of the powers to which the different prime numbers are raised to and multiplying them together (Notice that powers of 1 are not counted here as we are talking only about prime numbers and 1 is NOT a prime number). In this specific example, total number of factors= (4+1)*(2+1)= 15. Since 144 is a perfect square, its total number of factors is an odd number. You may try this with other numbers..

Hi,

Excellent point. You are exactly correct. The way to look at this is any number can be expressed as the product of primes raised to a power. So 144 = 2^4 * 3^2. All of the factors of 144 can be expressed as combinations of these primes raised to powers from 0 to 4 for 2, and 0 to 2 for 3.

In that case then the total number of combinations can be calculated as (4+1) * (2+1).

Nice job!

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Chris Brusznicki
MGMAT Instructor
Chicago, IL