We are asked to find the remainder when 1036823 is divided by 5. To solve this, we can use the properties of modular arithmetic.
We know that the remainder when a number is divided by 5 depends only on the last digit of that number. Therefore, we focus on the last digit of 1036823, which is 8.
Now, let’s find the remainder when 823 is divided by 5. We do this by looking at the pattern in the powers of 8 modulo 5:
We see that the powers of 8 modulo 5 repeat every 4 terms. Thus, we need to find the remainder when 23 is divided by 4.
\[ 23 \div 4 = 5 \, \text{remainder} \, 3 \]
This means that 8^23 mod 5 = 8^3 mod 5 = 2.
The remainder when 1036823 is divided by 5 is 2.
The correct answer is (b) 2.