Divisor counting problems

Last night my older son asked me for a little help understanding divisor counting problems. He was struggling a little bit with that type of problem at his math club.

Luckily Art of Problem Solivng’s Introduction to Number Theory book has an entire section on divisor counting problems, so it wasn’t difficult at all to find some good problems. We worked through four of them.

Divisor counting is one of my favorite topics – I remember learning about it in high school and being amazed that you could know the number of factors of an integer without listing all of them. That, all by itself, is a neat basic counting idea. Another reason that I like the topic is that it helps me sneak in a little arithmetic practice with the kids, and you’ll see why I like that practice in the first two videos.

Here are the four problems:

(1) Find the number of positive divisors of 999,999.


(2) What is the sum of the three positive numbers less than 1,000 that have exactly 5 positive divisors?


(3) If n has two prime divisors and 9 total divisors, how many divisors does n^2 have?


(4) How many divisors of 3,240 are (i) multiples of 3, and [separate question] (ii) how many are perfect squares?


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