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

$\begin{matrix} & 1 & \text{A} \ & \times & \text{A} \ \hline & 9 & \text{A} \ \hline \end{matrix}$

$\begin{matrix} & 1 & \text{A} \ & \times & \text{A} \ \hline & 9 & \text{A} \ \hline \end{matrix}$

Clearly, we will find the value of A.

A x A = A

⇒ A² = A

Find a number whose square gives the same number on unit place.

1 x 1 = 1

5 x 5 = 25

6 x 6 = 36

Lets take A = 1

$\begin{matrix} & 1 & \text{1} \ & \times & \text{1} \ \hline & 1 & \text{1} \ \hline \end{matrix}$

Hence, 1 does not satisfy the equation.

Lets take A = 5

$\begin{matrix} & \overset{2}{1} & \text{5} \ & \times & \text{5} \ \hline & 7 & \text{5} \ \hline \end{matrix}$

5 also does not satisfy the multiplication.

Lets take A = 6

$\begin{matrix} & \overset{3}{1} & \text{A} \ & \times & \text{A} \ \hline & 9 & \text{A} \ \hline \end{matrix}$

Hence, A = 6.