Math Problem Statement
can 3 to the power of 444 + 4 to the power of 333 be divide by 5
Solution
We are asked if is divisible by 5. To check this, we can use modular arithmetic.
Step-by-Step Solution:
We want to calculate , i.e., the remainder when is divided by 5.
Step 1: Find
Using Fermat's Little Theorem, we know that if is not divisible by , then: For , this tells us .
So, .
Step 2: Find
Similarly, we can use Fermat’s Little Theorem for as well. Since , we know:
Step 3: Add the results modulo 5
Now we compute the sum modulo 5:
Conclusion:
Yes, is divisible by 5.
Would you like more details or have any questions?
Here are 5 related questions to explore further:
- How does Fermat's Little Theorem help in modular arithmetic?
- Can we generalize this method for any prime divisor?
- What happens if we try dividing by numbers other than 5?
- How would the solution change if the exponents were different?
- Can this method be used for larger bases and powers efficiently?
Tip: Modular arithmetic often simplifies large exponentiation problems by reducing the problem into manageable cycles or periodic patterns.
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Math Problem Analysis
Mathematical Concepts
Number Theory
Modular Arithmetic
Exponentiation
Formulas
Fermat's Little Theorem: a^(p-1) ≡ 1 (mod p) for a prime p
Theorems
Fermat's Little Theorem
Suitable Grade Level
Grades 11-12