Digit Patterns of the Powers of 5

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

Hi, Tim. Interesting observation! The short answer to your question, 
"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.

In case you want to look up more information related to your 
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.

Here is something to think about that may help you explain the 
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 
another number instead of 5?

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 
is a lot more that I don't know about this subject.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/