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alexb
Charter Member
2190 posts 
Feb2308, 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 Theorem a^{p} ≡ 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 p a^{p}  a is even, so that, for any odd prime, a^{p} ≡ a (mod 2p) or, in case of p = 5, a^{p} ≡ a (mod 10) meaning, in part, that a and a^{p} end with the same digit. In the exceptional of a divisible by 5, its powers have the same last digit anyway. 

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Bui Quang Tuan
Member since Jun2307

Feb2308, 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,...9 May 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


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alexb
Charter Member
2190 posts 
Feb2308, 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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