Question

Given that the numbers 60AB57377 is divisible by 99, where A and B are digits. Find the possible value of A and B.

Answer 1

Digit will be 607357377 as a's value is 7 and b's value is 3 .

Answer 2

Solution :-

60AB57377 is divisible by 99.

So, it is also divisible by the factors of 99 which are 9 and 11.

Divisibility of 9
Sum of digits = 6+0+A+B+5+7+3+7+7
= A + B + 35
As the given number is divisible by 9, so A + B + 35 is also divisible by 9.
Therefore, possible values for A + B + 35 are 36, 45, 54, 63, 72, 81 .........
∴ A + B can be 1, 10, 19, 28, 37, 46, 55......   (1)

Divisibility of 11
Sum of the digits at even places = 0 + B + 7 + 7
= 14 + B
Sum of the digits at odd places = 6 + A + 5 + 3 + 7
= 21 + A

Difference between the sum = (21 + A) - (14 + B)
= 21 + A - 14 - B
= 7 + A - B

As the given number is divisible by 11,  7 + A - B is also divisible by 11.
∴ 7 + A - B can be 0, 11, 22, 33, 44 ...........
∴ A - B can be - 7, 4, 15, 26...... (2)

#) If A + B = 1; A = 0 or 1 and B = 1 or 0
∴ A - B
= 0 - 1
= - 1
Or
A - B 
= 1 - 0
= 1
Which do not satisfy the condition (2).

#) If A + B = 10, where A = 1, 2, 3, 4.........9 and B = 9, 8, 7, 6.........1
When A = 7 and B = 3,  
then A - B 
7 - 3 = 4  
This satisfies the condition (2).
Hence, the required values of A is 7 and B is 3

Let us check this by putting the possible values of A and B

607357377/99 = 6134923

So, 607357377 is exactly divisible by 99