If the nine-digit number 3422213AB is divisible by 99, then what is the value of 2A + B?

Given:

The nine-digit number 3422213AB is divisible by 99

Concept used:

Divisibility Rule of 9:

if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.

Divisibility Rule of 11:

If the difference of the sum of alternative digits of a number is divisible by 11, then the number itself is divisible by 11.

Calculation:

First, we check the divisibility with 9

⇒ 3 + 4 + 2 + 2 + 2 + 1 + 3 + A + B = 17 + A + B

The possible values of (A + B) are:

A + B = 1 ------- (1)

and A + B = 10 ------- (2)

Now, we check the divisibility with 11

⇒ (3 + 2 + 2 + 3 + B) - (4 + 2 + 1 + A) = 0

⇒ (10 + B) - (7 + A) = 0

⇒ A - B = 3 -------- (3)

And,

⇒ (3 + 2 + 2 + 3 + B) - (4 + 2 + 1 + A) = 11

⇒ (10 + B) - (7 + A) = 11

⇒ B - A = 8 ------------- (4) 

If we take equation (1) and (3) then;

A = (3 + 1)/2 = 2

B = (1 - 3)/2 = - 1

B = - 1 is not possible.

If we take equation (2) and (3) then;

A = (10 + 3)/2 = 13/2

Which is not possible. 

If we take equation (1) and (4) then;

B = (8 + 1)/2 = 9/2

Which is not possible.

If we take equation (2) and (4) then;

B = (10 + 8)/2 = 9

A = (10 - 8)/2 = 1

Putting the required value in the given equation:

⇒ 2A + B = 2 × 1 + 9 = 11

∴ The correct answer is 11.