# Divisibility

A couple weeks ago, I gave my Advanced Geometry class a problem on a problem set (unrelated to geometry):

I loved that many of my kids figured out the answer. It is $10^{24}$.

We then had a couple of “do nows” that related to this. So if you had 100!, what is the highest power of 7 that could go into it evenly? What about 11?

[After playing around with this puzzle, it is possible you will see why asking “What is the highest power of 2 that could go into it evenly” or “what is the highest power of 12 that could go into it evenly” are harder questions that 7, 11, or even 10.]

Kids were wondering if there was an efficient way to solve the problem. I said: yes. I won’t tell you what the way is, but I will introduce you to a math notation that might help… (and then I showed them the floor function).

Today a student in my Advanced Geometry class (who had seen the summation symbol and floor function in his middle school (!)) showed me this that he worked out:

Which is very close to being correct. It doesn’t work for composite numbers… But it does work if y is a prime number (or a power of a prime number). I pointed out to him that his solution wouldn’t work for the original Algebrainiac Attack problem, but it is very close to being correct. I asked him if he could modify his answers to deal with composite numbers… I am excited to see what he comes up with.

My one good thing? I gave my kids a concrete puzzle, and one kid wanted to generalize it! And he is getting very close to the solution!