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### Digit Patterns of the Powers of 5

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Date: 09/14/98 at 21:04:53
From: Tim Peterson
Subject: Multiplication digit patterns

I was playing with powers of 5, and I saw a pattern in the digits of
the results:

5^1:              5
5^2:             25
5^3:            125
5^4:            625
5^5:           3125
5^6:          15625
5^7:          78125
5^8:         390625
5^9:        1953125
5^10:       9765625
5^11:      48828125
5^12:     244140625
5^13:    1220703125
5^14:    6103515625
5^15:   30517578125

The 1's digit is always 5, and the 10's is always 2.
The third digit alternates between 1 and 6;
The fourth digit repeats 4 digits: 3, 5, 8, and 0;
The fifth digit repeats 8 digits: 1, 7, 9, 5, 6, 2, 4, and 0;
And the sixth (I think) repeats 16 digits.

What I want to know is why does each digit repeat an increasing power
of 2 number of digits?
```

```
Date: 09/16/98 at 12:57:53
From: Doctor Rick
Subject: Re: Multiplication digit patterns

"Why?" is, because 10 = 5 * 2. Of course, that calls for some further
explanation. I won't try to say everything that could be said on the
subject, but I'll point you to some areas for further investigation.

observations, they come under the branch of math called number theory,
and the topic in number theory called "residues of powers."

The first thing to notice is that, once a power of 5 has the same last
3 digits (for example) as a lower power of 5, the last 3 digits will
repeat forever. Can you see that from the way you multiply?

625     15625
* 5     *   5
----     -----
3125     78125

You start from the right, and by the time you have written the last 3
digits of the product, you haven't yet looked beyond the last 3 digits
of the multiplicand.

This means that, if 5^n and 5^(n+p) have the same last 3 digits, then
5^(n+1) and 5^(n+p+1) will have the same last 3 digits also, and so
on. The pattern repeats forever.

Now, there are "only" 1000 different numbers that you could have in
those last 3 digits, so sooner or later, as you keep computing powers,
you're bound to get a number that you already got for a lower power.
That means that if you raise any number to powers, you will find a
repeating pattern - only the repeat period is usually much longer.
The interesting question is, why does 5 give short repeat periods?

One thing you can do to expand your investigation is to look at
raising other numbers to powers. What are the repeat periods for the
last 1, 2, and 3 places? They are longer than for 5, so this will take
some work.

The next thing to think about is how this relates to modular
arithmetic. If two numbers (5^n and 5^(n+4), for instance) have the
same last 3 digits, this is the same as saying:

5^n mod 1000 = 5^(n+4) mod 1000

(Remember "mod", or "%" in the C language, means the remainder after
you divide.) In number theory, this is written as:

5^n is-congruent-to 5^(n+4) (mod 1000)

where is-congruent-to is an equal sign but with 3 lines instead of 2.
Another avenue for exploration is changing from 1000 to something
else. Try mod 2, mod 3, etc.

patterns you see. This relates to my comment about 10 = 5 * 2 being
important.

5^n       5^n      5^(n-3)
---- = ---------- = -------
1000   (2^3)(5^3)     2^3

Think about how the 3-digit pattern relates to mod 8, the 4-digit
pattern to mod 16, etc. Can the same sort of thing be done if you use

I have given you a lot more questions than answers. Have fun with it,
and be sure to let me know what you find. I have some ideas, but there

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory