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 is a prime number, and is any integer such that is not divisible by , then:
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.
- According to Fermat's Little Theorem, since is prime, we know: This statement is true.
2.
- We can rewrite this expression as: Using Fermat’s Little Theorem from the first statement, we know: So: Therefore, this statement is false.
3.
- From the second statement’s reasoning, we showed: This statement is true.
4.
-
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:
Would you like more details on any part of the solution?
Related Questions:
- Can Fermat’s Little Theorem be extended to composite numbers?
- How can we generalize Fermat’s Little Theorem to work with different bases?
- What is the Euler's theorem and how does it compare to Fermat's Little Theorem?
- How can we calculate modular exponentiation efficiently for large numbers?
- What happens when is divisible by 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)
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