Can you simplify a monstrous remainder problem using modular arithmetic?

In summary, the Hard Remainder Problem is a difficult mathematical problem that involves finding the remainder when dividing a large number by a smaller number. It is commonly used in cryptography to create secure algorithms, and cannot be solved using traditional methods. It is also related to the Chinese Remainder Theorem and has various applications in mathematics, computer science, and cryptography.
  • #1
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Find ${5^{2009}}^{1492}\mod{503}.$

How do you calculate a beast like this?
 
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  • #2
Do you know about Euler's theorem, or Fermat's little theorem? Powers of 5 are periodic modulo 503, so your expression can be simplified if you can find what that big exponent is modulo that period. Euler's theorem tells us that the period is divisible by divides 503 - 1 = 502 (since 503 is prime). Does that make sense?

If that doesn't help, what if you replaced 503 by, say, 7, does that make it simpler to reason about?
 
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FAQ: Can you simplify a monstrous remainder problem using modular arithmetic?

What is the Hard Remainder Problem?

The Hard Remainder Problem is a mathematical problem that involves finding the remainder when dividing a large number by a smaller number. It is known to be a difficult problem to solve and is often used in cryptography and computer science.

How is the Hard Remainder Problem used in cryptography?

The Hard Remainder Problem is used in cryptography to create secure algorithms for encryption and decryption. By using the difficulty of solving this problem, it helps to make the encryption process more secure and difficult to crack.

Can the Hard Remainder Problem be solved using traditional methods?

No, the Hard Remainder Problem cannot be solved using traditional methods such as long division. It requires more advanced mathematical techniques and algorithms to find the solution.

Is the Hard Remainder Problem related to the Chinese Remainder Theorem?

Yes, the Hard Remainder Problem is related to the Chinese Remainder Theorem, which is a mathematical theorem that helps to solve the Hard Remainder Problem in certain cases.

What are possible applications of the Hard Remainder Problem?

The Hard Remainder Problem has various applications in mathematics, computer science, and cryptography. It is used in creating secure algorithms, error-correcting codes, and in solving other mathematical problems such as the discrete logarithm problem.

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