Find the digits A and B, if :

$\begin{matrix} & \text{A} & \text{B} \ + & 3 & 7 \ \hline & 9 & \text{A} \ \hline \end{matrix}$

$\begin{matrix} & \text{A} & \text{B} \ + & 3 & 7 \ \hline & 9 & \text{A} \ \hline \end{matrix}$

Lets take A as a 1-digit number

B + 7 = A …..(1)

A + 3 = 9 …..(2)

A = 9 - 3 = 6

Putting the value of A in equation (1), we get

B + 7 = 6

B = 6 - 7

B = -1

∴ As we know B is a digit. Hence B ≠ -1.

Lets take A as a 2-digit number with ones digit as 1.

We can represent A as 10 + A

$\begin{matrix} & \overset{1}{\text{A}} & \text{B} \ + & 3 & 7 \ \hline & 9 & \text{A} \ \hline \end{matrix}$

B + 7 = 10 + A …..(3)

1 + A + 3 = 9 (one carry on ten's digit) …..(4)

A + 4 = 9

A = 9 - 4 = 5

Putting the value of A in equation 3.

B + 7 = 10 + 5

B + 7 = 15

B = 15 - 7 = 8

Hence, A = 5, B = 8

Now, lets take A as a 2-digit number with ones digit as 2.

We can represent A as 20 + A

$\begin{matrix} & \overset{2}{\text{A}} & \text{B} \ + & 3 & 7 \ \hline & 9 & \text{A} \ \hline \end{matrix}$

B + 7 = 20 + A …..(5)

2 + A + 3 = 9 (two carry on ten's digit) …..(6)

A + 5 = 9

A = 9 - 5 = 4

Putting the value of A in equation (5),

B + 7 = 20 + 4

B + 7 = 24

B = 24 - 7 = 17

∴ As we know B is a digit. Hence B ≠ 17.

$\begin{matrix} & \text{5} & \text{8} \ + & 3 & 7 \ \hline & 9 & \text{5} \ \hline \end{matrix}$

Hence, A = 5 , B = 8.