https://file.aiquickdraw.com/mathbot/file/1729252612262ozgqlrkf.jpg

We are tasked with determining whether the sum \( 3^{444} + 4^{333} \) 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.

\[
\begin{aligned}
3^1 &\equiv 3 \pmod{5}, \\
3^2 &\equiv 9 \equiv 4 \pmod{5}, \\
3^3 &\equiv 12 \equiv 2 \pmod{5}, \\
3^4 &\equiv 6 \equiv 1 \pmod{5}.
\end{aligned}
\]

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.

Is the number 3^444 + 4^333 divisible by 5?

Explore the divisibility of 3^444 + 4^333 by 5 using modular arithmetic. Learn about cyclic properties of powers under modulo operations. Suitable for high school students (Grades 9-12) to enhance understanding of number theory and modular arithmetic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation