Numer Properties Question

[rjcesaro]

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.

[jnelson0612]

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,

[thakurneelabh]

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.

[tim]

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..

[jnelson0612]

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.

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.

[thakurneelabh]

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.

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.

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..

[ChrisB]

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!

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..