Given: A number 1212121212... 300 times

To find: Remainder when the number is divided by 99

Solution:

Step 1: Observe the pattern of the given number

The given number is 12 repeated 300 times. We can write it as:

12 × (1010101010... 150 times)

Step 2: Divide the number inside the bracket by 99

To find the remainder when the number inside the bracket is divided by 99, we can use the following trick:

- Add the digits in odd places (1st, 3rd, 5th, etc.) and multiply by 11
- Add the digits in even places (2nd, 4th, 6th, etc.) and multiply by 9
- Find the difference between the two numbers obtained in steps 1 and 2

If the difference obtained in step 3 is divisible by 99, then the original number is also divisible by 99.

In this case, the number inside the bracket is 1010101010... 150 times. We can apply the above trick to find its remainder when divided by 99.

- Sum of digits in odd places = 1 + 1 + 1 + ... (150 times) = 150
- Sum of digits in even places = 0 + 0 + 0 + ... (150 times) = 0
- Difference = (11 × 150) - (9 × 0) = 1650

Step 3: Find the remainder when 12 × 1650 is divided by 99

We can simplify this as follows:

- 12 × 1650 = 12 × (99 × 16 + 6) = 1188 × 16 + 72
- 1188 is divisible by 99, so we only need to find the remainder when 72 is divided by 99

We can write 72 as 99 - 27, so:

- 72 ≡ -27 (mod 99)

Step 4: Combine the results from steps 2 and 3

The original number is:

12 × (1010101010... 150 times) ≡ 12 × 1650 (mod 99)

We found that 12 × 1650 is equivalent to -27 (mod 99), so:

12 × (1010101010... 150 times) ≡ -27 (mod 99)

Step 5: Find the positive remainder

We need to find the positive remainder when -27 is divided by 99. This can be written as follows:

- (-27) ÷ 99 = -1 with a remainder of 72
- Since we want the positive remainder, we add 99 to get:

- (-27) mod 99 = 72

Therefore, the remainder when 1212121212... 300 times is divided by 99 is 72.

Answer: (A) 18