# A B C in close knit relationship.. Who are they?

Find these closely knit numbers examining their relationships.

A, B, C are distinct positive integers. AB, AC are concatenated Numbers.

Given:

1 / AC = 0.0BC0BC0BC......

1 / BC = 0.0AC0AC0AC.......

If we try to solve the first equation

$$1 / AC = 0.\overline{0BC}~...~(1)$$
$$1000 / AC = BC.\overline{0BC}~...~(2)$$

$$(2) - (1) = 999 / AC = BC$$
$$999 = AC \times BC$$

For the second equation

It's the same as the previous one.

So

We need to solve $$999 = AC \times BC$$.
As the prime factorization of $$999$$ is $$3^3 \times 37$$ and we need both factors ($$AC$$ and $$BC$$) to have exactly $$2$$ digits, hence there is only one possible pair/answer which is $$27 \times 37$$.

Thus

$$A = 2, B = 3, C = 7$$ or $$A = 3, B = 2, C = 7$$