Introduction to Number Theory with my younger son

I saw this neat interview with Terry Tao yesterday:

In the first paragraph he mentions that he thinks that number theory isn’t likely to become an important subject in school math because it doesn’t have lots of applications.  I’m sure he is right, but agreeing with the idea doesn’t mean I have to like it!  I’m working through Art of Problem Solving’s “Introduction to Number Theory” book with my younger son this year and we are absolutely having a blast.   I’m obviously not suggesting a trip through Hardy and Wright, but the basic introduction to number theory in this book is so engaging, so fun and so useful for building up basic arithmetic skills, that I would happily suggest it for any kid looking to learn a little extra fun math.

The full talbe of contents is listed on the Art of Problem website here:

Go there for the chapters and subsections, but if you want a quick taste of the book the chapter titles are:

1. Integers:  The Basics

2. Primes and Composites

3. Multiples and Divisors

4. Prime Factorization

5. Divisor Problems

6. Special Numbers

7. Algebra with Integers

8.  Base Numbers

9. Base Number Arithmetic

10. Units Digits

11. Decimals and Fractions

12. Introduction to Modular Arithmetic

13. Divisibility Rules

14. Linear Congruences

15. Number Sense

I went through this book with my older son (also when he was in 3rd grade) and stopped after chapter 13.  I will probably stop at the same place here.

Maybe the Terry Tao interview from yesterday planted the seed in my mind, but the work we did this morning got me so excited that I wanted to write about it.  The problem we were tackling seemed pretty innocent to me at first:

Problem 4.7:

(a) Find the prime factorization of 45.

(b) Find the prime factorization of each of the four smallest multiples of 45 larger than 45:  90, 135, 180, and 225.

(c) What is the relationship between the prime factorizations from (b) and the prime factorizations from (a).

Yesterday we talked a little bit about factor trees and part (a) just reviews that topic.  He writes that 45 is 5 x 9 and then 5 x 3 x 3.    Not much to discuss, so we move to part (b).    I should say that he’s not writing out the products in the same way that I am here, he’s writing factor trees like the picture below.   Not sure how to format those trees in WordPress (or if taking the time to figure it out would improve the post!!).

Factor Trees

Next up  90 = 9 x 10 = 3 x 3 x 2 x 5.    I was expecting to see that he’d write 90 = 2 x 45, but I’m actually pretty happy to see that he didn’t think about this problem in that way.

135 = . . . . long pause.  Long, long pause, but he’s thinking so I don’t interrupt.  Suddenly he writes that 135 = 9 x 15 = 3 x 3 x 3 x 5.   I like the long think about factoring 135.  Hopefully that thinking is helping to build up a little number sense.

Next 180 = 10 x 18 = 2 x 5 x 2 x 9 = 2 x 5 x 2 x 3 x 3.

Now for the fun:

225 = long pause.

“Well, I know that 300 equals 15 x 20 and that 90 equals 15 x 6, so I know that 210 is 15 times something.”

Long pause.  Long pause and then he writes that 225 = 14 * 15.

“Are you sure?”


“Ok, l’ll tell you this – I don’t know what 14 x 15 is, but I know that it isn’t 225.  How could I know that?”

“Pause . . . . 14 is even and 14 x 15 has to be even. ”

“Interesting –  why don’t you multiply out 14 x 15 and see what it is.”

“[working it out]  210.”

“Good.  Remember that you said that 300 was 15 x 20 and that 90 was 15  x  6.  Do you see how to get to 210?”

“Yes, just subtract.”

“Great, now lets look at 225 again. ”

“225 = 15 x 15 = 3 x 5 x 3 x 5.”


Now, on to part (c) – what is the relationship between the factors above?  The goal here, I think,  is to notice that all of the numbers we factored in parts (a) and (b) could be written as 1 x 45, 2 x 45, 3 x 45, 4 x 45, and 5 x 45, but the way the numbers (or factors, I guess) were written on the board did not make that relationship obvious.    He thought about the question for a while and noticed that all of the numbers on the board had two 3’s and a 5 as factors.  It was neat to see him come to that conclusion and then eventually notice that what was left over was 1, 2, 3, 4, and 5.

So, a nice arithmetic review and a neat way to learn about factors and multiples all in one innocent litte problem.  He seems to really enjoy writing out the factor trees for various numbers – easy to forget how fun it is to learn ways to represent numbers that you’ve never seen before.  I also think that exercises like this are a great way to build number sense – so much thinking about multiplication in this problem.

As I said above, I’m a little sad to agree with the idea that number theory isn’t going to play much of a rule in a normal school math curriculum any time soon.  Maybe not every single kid is going to find exercises like this to be exciting, but I think that lots of kids will.   I’m sure enjoying walking through this book with my son.  Sort of sad to think that it is going to be my last time through it 😦