while multiplying 6 and B we get a number whose one's digit is B again

It is only possible when B = 0, 2, 4, 6, or 8

If B = 0, then the product will be 0.

Therefore, this value of B is immposible.

In that case,

If B = 2, then B × 6 = 12 and 1 will be a carry for the next step

6A + 1 = BB = 22

⇒ 6A = 21

Hence, any integer value of A is not possible

If B = 6, then B × 6 = 36 and 3 will be a carry for the next step

6A + 3 = BB = 66

⇒ 6A = 63

Hence, any integer value of A is not possible

If B = 8, then B × 6 = 48 and 4 will be a carry for the next step.

6A + 4 = BB = 88

⇒ 6A = 84

Therefore,

A = 14

However, A is a single digit number. Therefore, this value of A is not possible

If B = 4, then B × 6 = 24 and 2 will be a carry for the next step.

6A + 2 = BB = 44

⇒ 6A = 42

And hence, A = 7

The multiplication is as follows:

Hence, the values of A and B are 7 and 4 respectively