Question

Which of the following numbers is exactly divisible by 99?
A. 114345
B. 135792
C. 3572404
D. 913464

Answer 1

Hint:
We will first use the divisibility test of 9 on all the numbers until we get a number divisible by 9. If a number is found to be divisible by 9, we will then do divisibility test of 11 on that number, if that number passes the divisibility test of 11 too, We can say that the number is divisible by 11 and 9, Hence it is divisible by 99.

Complete step by step solution:
We know that according to the divisibility test of 9,
The sum of the digits must be equal to 9 or should be a multiple of 9.
We know that according to the divisibility test of 11,
The difference between the sum of the digits at even places and the sum of digits at odd places should be equal to 0 or should be a multiple of 11.
Now, according to the question, our first number is
114345
For divisibility test of 9, the sum of digits is
 $ \Rightarrow 1 + 1 + 4 + 3 + 4 + 5 = 18$
Since 18 is a multiple of 9, The given number is divisible by 9
For divisibility test of 11,
Sum of digits at even places
 $ \Rightarrow {S_{even}} = 1 + 4 + 4$
On simplification we get,
 $ \Rightarrow {S_{even}} = 9$
Sum of digits at odd places
 $ \Rightarrow {S_{odd}} = 1 + 3 + 5$
On simplification we get,
 $ \Rightarrow {S_{odd}} = 9$
The difference between the sum of the digits at even places and the sum of digits at odd places
 $ \Rightarrow D = {S_{even}} - {S_{odd}}$
On substituting the values we get,
 $ \Rightarrow D = 9 - 9$
On simplification we get,
 $ \Rightarrow D = 0$
Hence, the number is divisible by 11 also
Therefore, 114345 is the required number which is divisible by 99.

Hence, the correct answer is option A.

Note:
In these types of questions, we should stop after getting our required number, because in these questions only one of the options is correct, Hence to save time and efforts it is advisable to stop after finding the required number. But, if the question would have been asked as multiple correct questions, then we must proceed to check all the numbers.