Question

What will be remainder when $1212121212\dots 300$ times is being divided by 99?

• A. 18
• B. 81
• C. 54
• D. 36

Answer 1

The correct option is A 18

This number 1,212,121,212$\dots$ 300 times is divisible by 9. Therefore, we can write 1,212,121,212 $\dots$ 300 times = 9 N,
where N is the quotient obtained when divided by 9.
Now, this question is like $\frac{1212121212\dots 300times}{99}$
$=Remainder\frac{9}{9}\times \frac{134,680\dots written50times}{11}[121,212=13,468\times 9]$
Now we will have to find the remainder obtained when 134,680,134,680 -50 times is divided by 11. For this, we are supposed to use the divisibility rule of 11 from right-hand side.
[Using the divisibility rule from the left-hand side might give the incorrect remainder; similarly, if we find the remainder obtained when 12 is divided by I I, remainder $1=(2-1)\ne (1-2)$]
Remainder $\frac{134680\dots written50times}{11}=2$ Therefore, the total remainder = 18
Alternatively, divisibility rule of ${10}^n-1,n=2$ can be used to find the remainder in this case.