Crypto-arithmetic

Considering the ones column of the multiplication, the only way for $A \times 6$ to end in a digit of 0 is if $A = 5 .$ Therefore the equation is $55555 \times 6 = 333330 .$

Relevant wiki: Application of Divisibility Rules

We can rewrite the equation as $\overline{AAAAA} \times 6 = \overline{BBBBB0}$ . Since the left hand side of the equation is divisible by 6, then the right hand side is also divisible by 6. And so the right hand side is also divisible by 3 (and 2).

By divisibility rules of 3, if an integer is divisible by 3, then the sum of digits of the same integer is also divisible by 3,

$B + B + B + B + B + 0 = 5 B$ is divisible by 3. Since 5 is not divisible by 3, then $B$ must be divisible by 3.

Because $B$ represents a single digit integer, then $B$ can take the value 0, 3, 6 or 9 only.

If $B = 0$ , then $\overline{AAAAA} = 0 \div 6 = 0 \Rightarrow A = 0$ , but this is impossible because $A$ and $B$ are distinct digits.

If $B \geq 6$ , then $\overline{AAAAA} \leq 666660 \div 6 = 111110 > 100000$ is a 6-digit integer, which is absurd.

Hence, $B= \boxed3$ only, and $A = 5$ upon division.

A = 5. Only factor when multiplied by 6 that yields a product ending in 0; 6 X 5 =30,

6 X 55555 = 333330,

B = 3