Fifth grader Chas asks, "Is there only one way to create a factor tree, and why?"

Hi Chas, most numbers have more than one way to do a factor tree, but each factor tree gives the same prime factorization in the end. Let's take an example, so you can see what I mean. How would you like to do a factor tree for 72?

72 factors into 8 and 9, 8 factors into 4 and 2, 9 factors into 3 and 3, and 4 factors into 2 and 2. So the factor tree looks like this:

            72
          /     \
        8       9
      /   \     /  \
    4    2   3    3
   /  \ 
 2    2

So 72 factors into 2 x 2 x 2 x 3 x 3, or 2³3².

But there really isn't any reason that we started by splitting 72 into 8 and 9 (other than, perhaps, 8 x 9 = 72 is in the multiplication tables I memorized back when I was your age, so it was the first thing that popped into mind). Can you think of another pair of numbers that multiply to 72?  I thought of 36 and 2. So we could do the factor tree this way instead:

          72
         /    \
       36     2
     /      \
    6       6
  /   \     /   \
3    2   3     2

This factor tree looks very different from the first one, doesn't it? And yet, there are still 3 twos and 2 threes, giving us the same prime factorization: 2 x 2 x 2 x 3 x 3, or 2³3².

Is this always going to work? Yes! It doesn't matter how you split things up, and it doesn't matter how different your factor trees look. Every number has exactly one way of splitting it into prime factors, so the prime factorization will be the same no matter how you do the number's prime factorization.

And by the way, if you're nervous about doing a prime factorization wrong, this is a good way to check your work: do the factor tree, and then go back and do it a second time, only this time think of a different way of factoring the number. You'll have two different factor trees, but if you get the same prime factorization at the end, you can feel more confident that you got it right.

Of course, if you don't get the same factorization, you'll know you did something wrong, and you need to go back through your work to find out where you went wrong!

Have fun factoring, Chas!

Professor Puzzler