Question

The number which is exactly divisible by 99 is
[a] 3572404
[b] 135792
[c] 913464
[d] 114345

Answer 1

Hint: Use the fact that if p and q are primes, then a number is divisible by pq if and only if it is divisible by both p and q. Hence a number is divisible by 99 if and only if it is divisible by both 9 and 11. Use the divisibility rules of 9 and 11 to check whether a number is divisible by 9 and 11 and hence if it is divisible by 99.

Complete step-by-step answer:
Before solving the question, we need to know the divisibility rules of 9 and 11.
Divisibility by 9: A number is divisible by 9 if and only its sum of digits is divisible by 9.
Divisibility by 11: A number is divisible by 11 if the difference of the sum of the digits at even places and the sum of the digits at odd places is divisible by 11.
Now we check option wise divisibility by 9 and 11
Option [a] 3572404
Divisibility by 9:
Sum of digits $ =\text{ }3+5+7+2+4+0+4\text{ }=\text{ }25 $
Since 25 is not divisible by 9, 3572404 is also not divisible by 9
Divisibility by 99: Since 3572404 is not divisible by 9, 372404 is not divisible by 99.
Option [b] 135792
Divisibility by 9:
Sum of digits $ =\text{ }1+3+5+7+9+2=\text{ }27 $
Since 27 is divisible by 9, 135792 is also divisible by 9
Divisibility by 11:
Sum of digits at even places $ =\text{ }3+7+2\text{ }=12 $
Sum of digits at odd places $ =1+5+9=16 $
Difference $ =1612=4 $
Since the difference of the sum of the digits at even places and the sum of the digits at odd places is not divisible by 11, 135792 is also not divisible by 11.
Divisibility by 99: Since 135792 is not divisible by 11, 135792 is not divisible by 99.
Option [c] 913464
Divisibility by 9:
Sum of digits $ =9+1+3+4+6+4=27 $
Since 27 is divisible by 9, 913464 is also divisible by 9
Divisibility by 11:
Sum of digits at even places $ =1+4+4=9 $
Sum of digits at odd places $ =9+3+6=18 $
Difference $ =189=9 $
Since the difference of the sum of the digits at even places and the sum of the digits at odd places is not divisible by 11, 913464 is also not divisible by 11.
Divisibility by 99: Since 913464 is not divisible by 11, 913464 is not divisible by 99.
Option [d] 114345
Divisibility by 9:
Sum of digits $ =1+1+4+3+4+5=18 $
Since 18 is divisible by 9, 114345 is also divisible by 9
Divisibility by 11:
Sum of digits at even places $ =\text{ }1\text{ }+\text{ }3\text{ }+\text{ }5\text{ }=9 $
Sum of digits at odd places $ =\text{ }1+\text{ }4\text{ }+4\text{ }=9 $
Difference $ =\text{ }9\text{ }9\text{ }=\text{ }0 $
Since the difference of the sum of the digits at even places and the sum of the digits at odd places is divisible by 11, 114345 is also divisible by 11.
Divisibility by 99: Since 114345 is divisible by both 9 and 11, 114345 is also divisible by 99.
Hence option [d] is correct.

Note: Verification:
We have $ 3572404=99\times 36084+88 $
Hence, we have 3572404 is not divisible by 99
We have $ 135792=99\times 1371+63 $
Hence, we have 135792 is not divisible by 99
We have $ 913464=99\times 9226+90 $
Hence, we have 913464 is not divisible by 99
We have $ 114345=99\times 1155 $
Hence, we have 114345 is divisible by 99.
Hence our answer is verified to be correct.