Even numbers such as 8 have an even number of factors {1,2,4,8} and odd numbers such as 9 have an odd number of factors {1,3,9}, right?

1. May 15, 2006 am31 6:15 am 6:15 am

But 15, an odd number, has 4 factors {1,3,5,15}. If there are no squares among a number’s factors, there’s an even number of factors.

If there’s a square among the factors, things get odd, right?

2. May 15, 2006 am31 11:20 am 11:20 am

John, what about 24, which has a square among its factors, yet has 8 factors: {1,2,3,4,6,8,12,24}?

3. May 15, 2006 pm31 9:38 pm 9:38 pm

Well, the theme or meme for this post seems to be that the answer to the “right” at the end of the sentence is “not quite.”

So, okay, it’s not quite right.

As you seem to know, the number of factors can be found by taking the prime factorization of a number, adding 1 to each exponent, and then multiplying the augmented exponents.

That’s clear as mud. Let’s take an example.

24 = 2EXP3 X 3. The exponents are 3 and 1. Adding 1 to each gives us 4 and 2. Multiplying gives us 8, which is the number of factors.

So, yes, if there’s not just a square but a cube involved, you can end up with an even number of factors.

4. May 15, 2006 pm31 10:26 pm 10:26 pm

“As you seem to know, the number of factors can be found by taking the prime factorization of a number, adding 1 to each exponent, and then multiplying the augmented exponents. ”

Ah, there goes my next puzzle post. I love that point. Algebra makes the arithmetic clearer (I think)

Consider ab^3 ( b^3 for “b to the third power”)

the factors are:
1 a
b ab
b^2 ab^2
b^3 ab^3

I am sorry that I don’t know how to make tables yet, but there is your (1 + 3) x (1 + 1) table, not as clear as day, but clearer than mud.

And of course, this applies just fine to 24 (a=3, b=2)

5. May 17, 2006 am31 1:01 am 1:01 am

Another approach to this puzzle…

The factors of a number n can be written in pairs: f is a factor iff n/f is a factor.

Hence every number has an even number of factors except when (for one of the factors) f and n/f coincide, in which case n is a perfect square.

6. May 16, 2010 pm31 8:50 pm 8:50 pm

Did you know this has a “Practical” application. Take 100 lights all turned off, numbered 1..100. Toggle all the switches. Toggle all the even ones. Toggle 3, 6, 9, etc. Toggle 4, 8, 12, etc. All the way up to toggling all those divisible by 100.

What lights are on now?

• May 16, 2010 pm31 11:54 pm 11:54 pm

Interesting, but I have to point out, hardly practical.