Question

What will be the remainder left when 10368²³ is divided by 5?

• A. 4
• B. 2
• C. 3
• D. 1

Answer 1

The Correct Option is B

Remainder Calculation Using Modular Arithmetic 

Step 1: Focus on the Last Digit of the Number

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.

Step 2: Find the Remainder When 823 is Divided by 5

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:

• 8^1 mod 5 = 8 mod 5 = 3
• 8^2 mod 5 = 64 mod 5 = 4
• 8^3 mod 5 = 512 mod 5 = 2
• 8^4 mod 5 = 4096 mod 5 = 1

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.

Step 3: 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.

Final Answer:

The remainder when 1036823 is divided by 5 is 2.

Conclusion:

The correct answer is (b) 2.