Math Problem Statement
Is the number 3^444 + 4^333 divisible by 5?
Solution
We are tasked with determining whether the sum is divisible by 5. To solve this, we can use modular arithmetic to analyze the problem modulo 5.
Step 1: Find the pattern of powers of 3 modulo 5
We calculate the powers of 3 modulo 5 and observe any repeating pattern.
3^1 &\equiv 3 \pmod{5}, \\ 3^2 &\equiv 9 \equiv 4 \pmod{5}, \\ 3^3 &\equiv 12 \equiv 2 \pmod{5}, \\ 3^4 &\equiv 6 \equiv 1 \pmod{5}. \end{aligned}$$ Since $$3^4 \equiv 1 \pmod{5}$$, the powers of 3 repeat every 4 terms. To find $$3^{444} \mod 5$$, we compute the remainder of $$444 \div 4$$, which gives $$444 \equiv 0 \pmod{4}$$. Hence, $$3^{444} \equiv 3^0 \equiv 1 \pmod{5}.$$ ### Step 2: Find the pattern of powers of 4 modulo 5 Similarly, we calculate the powers of 4 modulo 5. $$\begin{aligned} 4^1 &\equiv 4 \pmod{5}, \\ 4^2 &\equiv 16 \equiv 1 \pmod{5}. \end{aligned}$$ Since $$4^2 \equiv 1 \pmod{5}$$, the powers of 4 repeat every 2 terms. To find $$4^{333} \mod 5$$, we compute the remainder of $$333 \div 2$$, which gives $$333 \equiv 1 \pmod{2}$$. Hence, $$4^{333} \equiv 4^1 \equiv 4 \pmod{5}.$$ ### Step 3: Sum the results modulo 5 Now we can sum the results: $$3^{444} + 4^{333} \equiv 1 + 4 \equiv 5 \equiv 0 \pmod{5}.$$ ### Conclusion: The number $$3^{444} + 4^{333}$$ is divisible by 5. Do you need further clarification or details on any part of the solution? Here are 5 related questions to deepen understanding: 1. What are the steps to calculate powers modulo a number? 2. How can we apply modular arithmetic to other divisibility problems? 3. Why do powers of numbers often exhibit cycles when taken modulo a number? 4. How can modular arithmetic simplify computations of very large powers? 5. Can we extend this method to check divisibility by other numbers (e.g., 7 or 11)? **Tip**: When dealing with large powers and divisibility, recognizing patterns through modular arithmetic can greatly simplify the problem.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Exponentiation
Divisibility
Formulas
Modulus (a ≡ b mod m)
Powers of integers modulo a number
Theorems
Modulo Arithmetic Cyclic Properties
Suitable Grade Level
Grades 9-12