A neat “last digit” problem from James Tanton

My younger son and I finished up the “last digit” chapter in our Introduction to Number Theory book today, and I was looking for a fun last exercise. Luckily James Tanton had my back this weekend:

I decided to use this problem as tonight’s exercise and even asked my older son to join in since it looked like we’d have a nice discussion.

We started by just talking through the problem a little bit. I asked each kid to rank the four problems from what they thought would be the easiest to what they thought would be the hardest. It was interesting that their ranking was not the same.

My younger son started off on what he thought was the easiest problem – the pattern in the last digit of the powers of 3:


Next up was the powers of 5. My older son recognized that this would be an easy problem. We talked for a little bit about why the last digit was always 5 and managed to squeeze out a pretty good (if short) discussion. After that we moved on to talking about the harder problems. Neither kid had any ideas about how to proceed with the larger numbers. Eventually my younger son suggested that it would be a good idea to move to the computer for the more difficult problems, so that’s what we did.


We ended the last movie trying to guess how long it would take the powers of 13 to repeat the last two digits. Both kids thought the repetition would happen pretty quickly – about every 2 or every 4 times. We’ve mostly been talking about last digits, so I guess that conditions you to believe that repetitions will happen pretty frequently.

We saw immediately that the repetition didn’t occur quickly. My younger son realized that we should be looking for a last two digits of 01 to singal the repetition – I was happy to hear that since that showed he was starting to understand what to look for.

At the end of this movie we were talking about moving on to look at 113. My older son thought there might be a pattern going from 3 to 13 to 113. Love when they have ideas like that.


The last thing we did was look at the last three digits of powers of 113. It turns out that my son’s guess for the length of the repeating pattern was right. He was pretty excited about that! To see if his patterned continued we looked to see if the last four digits of 1113 cycled every 500 powers.

In all honesty, I don’t know why this pattern is happening, but the kids really liked it. Even if it was just a guess, it was neat to see how happy my son was that he’d guessed the pattern correctly.


So, another really fun math conversation with the kids from a James Tanton tweet. I’ve really enjoyed the section in our number theory book on last digits – it seems to have really helped my younger son build on his existing number sense. The problems with the last two, or three, or more digits are fun, too. They also are great, and relatively straightforward, ways to introduce a little computer math to kids. Definitely a fun little conversation tonight.

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