What is the remainder when 3^444 + 4^333 is divided by 5?

Sure, let's solve this step by step:

Step 1: We need to find the remainder of 3^444 and 4^333 when divided by 5.

Step 2: Notice that 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, and 3^5 = 243. If you look at the last digit of each of these numbers, you'll see a pattern: 3, 9, 7, 1, 3, 9, 7, 1, and so on. This pattern repeats every 4 powers. So, the last digit of 3^444 will be the same as the last digit of 3^4, which is 1.

Step 3: Similarly, for 4^333, the pattern of the last digit is: 4, 6, 4, 6, and so on. This pattern repeats every 2 powers. So, the last digit of 4^333 will be the same as the last digit of 4^1, which is 4.

Step 4: Now, add these two numbers together: 1 (from 3^444) + 4 (from 4^333) = 5.

Step 5: The remainder when 5 is divided by 5 is 0.

So, the remainder when 3^444 + 4^333 is divided by 5 is 0.