Math Problem Statement

Assume that 6473 is a prime. Which of these statements are true according to Fermat's little theorem

3^6472 mod 6473 = 1 3^6473 mod 6473 = 1 3^6473 mod 6473 = 3 3^6472 mod 6472 = 1

Solution

Let's analyze each statement in the context of Fermat's Little Theorem, which states:

  • If pp is a prime number, and aa is any integer such that aa is not divisible by pp, then: ap11(modp)a^{p-1} \equiv 1 \pmod{p}

Given that 6473 is a prime number, and assuming 3 is not divisible by 6473 (which it isn't), we can apply Fermat's Little Theorem to evaluate the truth of each statement.

1. 36472mod6473=13^{6472} \mod 6473 = 1

  • According to Fermat's Little Theorem, since 64736473 is prime, we know: 364721(mod6473)3^{6472} \equiv 1 \pmod{6473} This statement is true.

2. 36473mod6473=13^{6473} \mod 6473 = 1

  • We can rewrite this expression as: 36473=3×364723^{6473} = 3 \times 3^{6472} Using Fermat’s Little Theorem from the first statement, we know: 364721(mod6473)3^{6472} \equiv 1 \pmod{6473} So: 364733×1=3(mod6473)3^{6473} \equiv 3 \times 1 = 3 \pmod{6473} Therefore, this statement is false.

3. 36473mod6473=33^{6473} \mod 6473 = 3

  • From the second statement’s reasoning, we showed: 364733(mod6473)3^{6473} \equiv 3 \pmod{6473} This statement is true.

4. 36472mod6472=13^{6472} \mod 6472 = 1

  • Fermat's Little Theorem applies when the modulus is a prime number, but here the modulus is 6472, which is not prime. Therefore, we cannot use Fermat's Little Theorem directly in this case.

    This statement is false.

Summary:

  • The true statements are:
    • 36472mod6473=13^{6472} \mod 6473 = 1
    • 36473mod6473=33^{6473} \mod 6473 = 3

Would you like more details on any part of the solution?


Related Questions:

  1. Can Fermat’s Little Theorem be extended to composite numbers?
  2. How can we generalize Fermat’s Little Theorem to work with different bases?
  3. What is the Euler's theorem and how does it compare to Fermat's Little Theorem?
  4. How can we calculate modular exponentiation efficiently for large numbers?
  5. What happens when aa is divisible by pp in Fermat's Little Theorem?

Tip: Modular exponentiation can be efficiently calculated using a method called exponentiation by squaring, which reduces the number of multiplications needed.

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Math Problem Analysis

Mathematical Concepts

Number Theory
Modular Arithmetic
Prime Numbers

Formulas

a^(p-1) ≡ 1 (mod p) when p is prime and a is not divisible by p

Theorems

Fermat's Little Theorem

Suitable Grade Level

Undergraduate Level (First-year Mathematics or Discrete Mathematics Course)