My younger son and I are going through the section in Art of Problem Solving’s Introduction to Number theory book about linear congruences. This is a pretty advanced topic. I actually skipped it with my older son, but my younger son absolutely loves it. He’s totally captivated and I love hearing his ideas about how to solve the problems.

Today we were discussing how you solve linear equations using modular arithmetic. In particular, which equations have solutions and which ones don’t. Again, a pretty advanced topic for kids – and, to be clear, not one that I plan on covering in a lot of depth – but talking through this with him was so fun. It is also an area of math where knowing the definition of division as multiplication by the reciprocal is useful.

We started off talking about an equation that has a solution. I really enjoyed hearing him reason his way through this problem – especially the part where he pauses to make sure that 3 has an inverse.

Next we talked through an example where there is no solution. One interesting part of this discussion for me is his body language. You can see that the equation makes him uncomfortable – there’s no solution – but he struggles for a bit to figure out how to put this idea into words. Then he just crosses out the whole equation – ha!

Finally an example where the solution is slightly harder to get to than in the first example. Here we have a common factor between three parts of the equation. That fact allows us to modify the equation to find a solution. A complete understanding of the ideas here is likely a little bit beyond his understanding right now, and definitely beyond what I’m trying to teach him. It is still pretty interesting to me to hear him work through the problem here.

So, a tough, but fun topic. I’m thankful that I have the chance to spend a little extra time with my son working through this section. It makes me really happy to see him having so much fun with the math here.