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/
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