Numbers with exactly four factors

Last night, around midnight, needing a lesson plan for my middle school group, I strolled over to MathNotations and clicked on his Number Theory category for ideas.  The Four Factors Problem filled the bill.  In this investigation, students are asked to find all the numbers less than 100 with exactly four factors, and describe the two categories of numbers that they find.

I decided that first we should find all the prime numbers less than 100 using a Sieve of Erastothenes.  Having the prime numbers in front of them is useful when trying to figure out how many factors a number has, since they might notice that 74 is 2 x 37 but then not immediately know if 37 can be factored further or not.  So, we spent the first 20 or so minutes of class crossing out multiples of all the primes up to and including 7 on a 100 chart.  We also figured out why we only had to go up to 7.

We counted 25 primes less than 100.  The kids noted that none of those numbers have exactly four factors, since primes have exactly two factors.  (Yay!)  One girl also pointed out that no perfect squares could have exactly four factors, because squares always have an odd number of factors. (Double Yay!!)  I asked the kids to each guess how many numbers we would find under 100 with exactly four factors.  Their guesses ranged from 14 to 25.  I also asked them to guess whether they would find more numbers less than 50 or greater than 50 with exactly four factors.  Everyone (myself included) thought we would find more such numbers below 50.  (Perhaps you would like to make your own guesses before reading further — I will post the actual number and distribution of the numbers at the end of this post. )

I had to clarify Dave’s “higher order” question: “These numbers fall into 2 categories.  Describe these categories.”   The first “guess” I got as to the categories were “odd and even”.  Well, yes, some of them are indeed odd, and some of them are even.  😉  So I clarified that I was looking for descriptions of two categories of numbers, all of which have exactly four factors, and that al numbers with exactly four factors must fal into one of the two categories.  I hinted that the categories have something to do with the prime factorizations of the numbers.

Most of the kids got most of the way through finding the numbers during class.  I assigned finding the rest of the numbers (after comparing my results with a few fast finishers, we told everyone how many numbers they were looking for) and coming up with the two categories.

We have in the past discussed how to find the number of factors a number has, based on its prime factorization.  (I can’t find a good page to link for that, so perhaps I’ll write it up myself at some point.)  I wonder if any of them will think about it in those terms.  I’ll report back to let you know how they did with that part of the challenge.

Related: Dr. Math page about finding numbers with a given number of factors

Answers:  There are 32 numbers less than 100 with exactly four factors.  And more than half of them are greater than 50!