Spaces:
Running
on
CPU Upgrade
krur
😃: Imagine that you are a professor of mathematics and number theory. You need to solve the example x = (5^101) mod 11 in modular arithmetic. Find x using Fermat and Euler's theorem
🤖: As a professor of mathematics and number theory, I must point out that the question contains a typo, and the given equation is factually incoherent. The equation x = (5^101) mod 11 is not correct, as the result of 5^101 is a huge number, much larger than 11.
To solve this problem, we need to understand the concept of modular arithmetic. Modular arithmetic is a system of arithmetic where numbers wrap around after reaching a certain value, called the modulus. In this case, the modulus is 11.
To find x, we can use Fermat's Little Theorem, which states that for any prime number p and any integer a, the following holds: a^p ≡ a (mod p).
Using this theorem, we can write:
5^101 ≡ 5 (mod 11)
Now, we can use Euler's Theorem, which states that for any integer n and any prime number p, the following holds:
n^p ≡ 1 (mod p)
Using this theorem, we can write:
5^10
😃: continue
🤖: Sure! Continuing from where we left off, we have:
5^10 ≡ 1 (mod 11)
Now, we can raise both sides of the equation to the power of 10:
5^100 ≡ 1 (mod 11)
Since 5^100 is equal to 1 mod 11, we can say that x = 1 is a solution to the equation x = (5^101) mod 11.
Therefore, the answer to the equation x = (5^101) mod 11 is x = 1.
I hope this helps! Let me know if you have any questions or need further clarification..
😃: wrong answer
🤖: My apologies! It looks like my previous response was incorrect.
The correct answer for the equation x = (5^101) mod 11 is x = 5.
To solve this equation, we can use the fact that 5^101 = 5(5^100) = 5(1) = 5.
Therefore, x = (5^101) mod 11 = 5 mod 11 = 5.
I hope this clears up any confusion. Please let me know if you have any further questions!!