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 Subject: "Fifth Power Of Digits" Previous Topic | Next Topic
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Bui Quang Tuan
Member since Jun-23-07
Feb-22-08, 00:02 AM (EST)    "Fifth Power Of Digits"

 Dear All My Friends,Why fifth power of any digit is ended with this same digit also?Here is my calculations:0^5 = 01^5 = 12^5 = 323^5 = 2434^5 = 10245^5 = 31256^5 = 77767^5 =168078^5 = 327689^5 = 59049Thank you and best regards,Bui Quang Tuan

alexb
Charter Member
2190 posts
Feb-23-08, 00:11 AM (EST)    1. "RE: Fifth Power Of Digits"
In response to message #0

 >Why fifth power of any digit is ended with this same digit >also? According to Fermat's Little Theoremap ≡ a (mod p)for any a not divisible by p, p a prime.Well, 5 is a prime and, for any a, and for any pap - a is even,so that, for any odd prime,ap ≡ a (mod 2p)or, in case of p = 5,ap ≡ a (mod 10)meaning, in part, that a and ap end with the same digit.In the exceptional of a divisible by 5, its powers have the same last digit anyway. Bui Quang Tuan
Member since Jun-23-07
Feb-23-08, 07:06 AM (EST)    2. "RE: Fifth Power Of Digits"
In response to message #1

 Thank you for interesting proof by using interesting theorem. I have found two following proofs:1. y = x^5 - x = (x - 1)*x*(x + 1)*(x^2 + 1)Therefore y = 0 (mod 5) and y = 0 (mod 2). It means x^5 and x are ended with the same digit.2. (x + 1)^5 = (x^5 + 5*x^4 + 10*x^3 + 10*x^2 + 5*x + 1) = 5*x*(1 + x)*(1 + x + x^2) + (x^5 + 1)Easy to show 5*x*(1 + x)*(1 + x + x^2) = 0 (mod 10)Therefore the last digit of (x + 1)^5 is the last digit of x^5 plus 1. Start with x = 0 we can get all the last digits of 0, 1, 2, ... 9 are the same 0, 1, 2,...9May be exist also another interesting proofs?I am thinking now about two small questions:1. Why after one, five is the first digit with this property?2. Why other one, only 5 and 6 hold the property: the last digit of x^n is the same x with any n?Best regards,Bui Quang Tuan alexb
Charter Member
2190 posts
Feb-23-08, 04:46 PM (EST)    4. "RE: Fifth Power Of Digits"
In response to message #2

>1. Why after one, five is the first digit with this
>property?

Because 10 = 2*5 and 2 is too small.

>2. Why other one, only 5 and 6 hold the property: the last
>digit of x^n is the same x with any n?

Base
6
3² = 13

4² = 24
10
5² = 25

6² = 36
12
4² = 14

9² = 69
14
7² = 37

8² = 48
15
6² = 26

A² = 6A
18
9² = 49

A² = 5A
21
7² = 27

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