In An exceptional talent for calculative thinking, (IML Hunter, 1962), Professor Aitken explained how he found the decimal expansion of 1/851:
851 is 23 times 37. I use this fact as follows. 1/37 is 0.027027027027. . .. This I divide mentally by 23... 1/37 recurs at three places, 1/23 recurs at twenty-two places, the lowest common multiple of 3 and 22 is 66, whence I know that there is a recurring period of 66 places.’
I thought this is a general fact: two rationals with decimal periods a and b must have a decimal period of ab, but there is a clear counterexample:
$$0.\dot 2 \times 0.\dot 1 \dot 5 = 0.\dot{0}3367\dot{0}$$
So what is the general law?
A rule is that the product of two repeating decimals has period the least common multiple of their periods provided the denominators are relatively prime and the numerator of each fraction is relatively prime to the denominator of the other.