# Periodic Last Digits of Fibonacci Numbers

Here’s something I was playing around with the other day.

Consider the Fibonacci sequence: $0, 1, 1, 2,3,5,8,13,21, \dots$. $F(n) = F(n-1)+F(n-2)$. I wanted to see if there was a pattern regarding the values of the ones digit in the fibonacci sequence. This is what I found:

The ones digits of the Fibonacci numbers are periodic, with period 60. This means that if you continue the sequence and look only at the ones digits of the numbers, they begin to repeat the same pattern after the 60th number. In addition, every 15th number in the sequence has a ones digit of zero. I wonder why this is. I can’t seem to find any other patterns at the moment. It probably has something to do with mod.

Here is the list of the 60 digit period (ones digits only):

```1
1
2
3
5
8
3
1
4
5
9
4
3
7
0
7
7
4
1
5
6
1
7
8
5
3
8
1
9
0
9
9
8
7
5
2
7
9
6
5
1
6
7
3
0
3
3
6
9
5
4
9
3
2
5
7
2
9
1
0
---this is where the cycle starts over
1
1
2
3
5
8
...```

Can you find any other interesting properties?

## 7 comments on “Periodic Last Digits of Fibonacci Numbers”

Did you try to do this calculation for mod 100? Don’t believe it’s just coincidence 🙂

• That’s definitely something to look into, I’ll check it out, thanks for the tip!

• anton says:

with mod 100, it repeats every 300th nr

2. anubhav says:

What about mod with a prime number, say 10^9 +7 …

3. abean077 says:

This isn’t really a property unique to the Fibonacci sequence, or very special. Any sequence for which a term of the sequence depends only on a finite amount of previous terms, like the Fibonacci sequence depends only on the two before, and powers of 2 depend only on the one before, repeats like this, even if you use a different base. Why? Consider a sequence that depends only on some number of the last terms. Then there is only a finite amount of possible last-digit configurations, 100 for the Fibonaccis, 10 for the powers of 3, 1000 for a Fibonacci-like sequence with adding the last three terms instead. So at some point, since the sequence goes on infinitely, the same configuration must repeat, by Pigeonhole. If the same configuration repeats multiple times, the number following the repeated configuration will also be the same, and the sequence repeats.

For example, powers of 2 go 2, 4, 8, 6 and repeat. 3’s go 3, 9, 7, 1.

4. brianforst says:

It repeats in the units digit after 60; in the tens digit after 300; and in the hundreds digit after 1500. Can you prove that it will eventually repeat in every digit?

5. Shubham Goel says:

no after hundreds digit i.e. thousands digit,it repeat after 15000 and ten thousands digit after 150000..so you can’t prove it will eventually repeat in every digit