In each of the following cases, find the least value/values of letters used in place of digits :

$\begin{matrix} & \text{A} & 1 \ + & 1 & \text{B} \ \hline & \text{B} & 0 \ \hline \end{matrix}$

$\begin{matrix} & \text{A} & 1 \ + & 1 & \text{B} \ \hline & \text{B} & 0 \ \hline \end{matrix}$

Firstly we will find the value of B.

Clearly, 1 + B is a number whose ones digit is 0.

⇒ 1 + B = 0, 1 + B = 10, 1 + B = 20; and so on.

⇒ B = 0 - 1, B = 10 - 1, B = 20 - 1; and so on.

⇒ B = -1, B = 9, B = 19; and so on.

Since, B is a digit, ∴ B = 9.

Secondly we will find the value of A.

$\begin{matrix} & \overset{1}{\text{A}} & 1 \ + & 1 & \text{9} \ \hline & \text{9} & 0 \ \hline \end{matrix}$

Clearly, 1 is carried over on tens place. A + 1 + 1 = 9.

⇒ A + 2 = 9 , A + 2 = 19 , A + 2 = 29; and so on.

⇒ A = 9 - 2 , A = 19 - 2 , A = 29 - 2; and so on.

⇒ A = 7 , B = 17 ,B = 27; and so on.

Since, A is a digit.∴ A = 7.

$\begin{matrix} & \text{7} & 1 \ + & 1 & \text{9} \ \hline & \text{9} & 0 \ \hline \end{matrix}$ A = 7 and B = 9