10-adic number proof: A^10 has the same n+1 last digits as 1

In summary, a 10-adic number proof is a mathematical proof that uses 10-adic numbers to show relationships between numbers. The statement "A^10 has the same n+1 last digits as 1" means that when A is raised to the power of 10, the resulting number will have the same last n+1 digits as 1. This proof works by representing the resulting number as a sum of multiples of powers of 10 and examining the last n+1 digits. The significance of this proof lies in its implications in number theory and cryptography. It has real-life applications in public key cryptography, digital signatures, and error-correcting codes.
  • #1
tomkoolen
40
1
The question at hand: Let X be a 10-adic number. Let n be a natural number (not 0). Show that A^10 has the same n+1 last digits as 1 if A has the same n last digits as 1 (notation: A =[n]= 1)

My work so far:

(1-X)^10 = (1-X)(1+X+X^2+...+X^10)
A =[n]= 1
1-A =[n]= 0.
I think I can also say that 1+X+X^2+...X^10 =[n]= 1.
But now I don't know how to continue. Could anybody help me out?

Thanks in advance.
 
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  • #2
Of course A = X, my bad!
 

Related to 10-adic number proof: A^10 has the same n+1 last digits as 1

1. What is a 10-adic number proof?

A 10-adic number proof is a mathematical proof that uses the concept of 10-adic numbers, which are a way of representing numbers in base 10 but with infinitely many digits to the left of the decimal point. This type of proof is often used in number theory to show relationships between numbers.

2. What does the statement "A^10 has the same n+1 last digits as 1" mean?

This statement means that when the number A is raised to the power of 10, the resulting number will have the same last n+1 digits as the number 1. This is a specific case of a more general proof that shows that A^n will have the same n+1 last digits as 1 for any positive integer n.

3. How does this proof work?

This proof uses the concept of 10-adic numbers to show that when A is raised to the power of 10, the resulting number can be written as a sum of multiples of powers of 10. By examining the last n+1 digits of this sum, it can be shown that they are the same as the corresponding digits of the number 1, proving the statement to be true.

4. What is the significance of this proof?

This proof may seem like a trivial statement, but it has important implications in number theory and cryptography. It can be used to prove the existence of numbers with specific properties and to show relationships between different numbers. In cryptography, it can be used to create secure encryption methods.

5. Are there any real-life applications of this proof?

Yes, there are many real-life applications of this proof. One example is in the field of public key cryptography, where it is used to generate secure encryption keys. It is also used in digital signatures, error-correcting codes, and in various other areas of mathematics and computer science.

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