# Questions from middle schoolers VII: Proper factors

### Why do we talk about “proper factors” but not “proper multiples”?

Good question.

I had never thought about this before. A colleague and I are thinking about this. My colleague’s guess is that in number theory (the mathematical field that deals with this stuff), there are some things that are true of proper factors that are not true of the number itself, and that this is not a problem with multiples.

But I don’t really know for sure. I’ll find out.

### 3 Responses to Questions from middle schoolers VII: Proper factors

1. The most I’ve seen here is how “proper factors” allow you to give more intuitive definitions of abundant, deficient, and perfect numbers.

The only thing proper factors have going for them is being less than the number (while still positive). Other than the perfect number (1+2+3 = 6 when the sum of the proper factors equals the number) it seems to be a pretty useless term, and therefore should get the shaft!

Most number theory applications use the total number of factors, not the number of proper factors. An exploration on the number of factors of different numbers gets interesting in a hurry — there are 3 factors of 4 and 4 factors of 15… how many factors of 60 are there?

- Bowen

2. Christopher

I was thinking about this today, remembering that I never got back to the middle schoolers on this topic. I appreciate your contribution, Bowen.

I will add something implicit in your response-that a number’s factors are finite in quantity, while a number’s multiples are infinite. Therefore, we spend time counting the factors of a number, and no time at all counting its multiples.

What I love about this question, though, is its exploratory, probing nature. There is a raw and lovely curiosity behind this question. These students are making connections and they’re not taking adults’ BS answers at face value. It’s great stuff, I think.

3. Christopher

Now, almost a year later, I want to make a pitch for proper multiples. Through working on an A Assignment, my future elementary teachers have developed a theorem:

All proper multiples of perfect numbers are abundant.

Proper is useful here because the perfect number itself is not abundant, but the rest of its multiples are.

So there you have it. We don’t talk about proper multiples because we don’t need the term. Now we have a theorem that the term is useful for. So now we’ll talk about proper multiples.