Define and . Then . Therefore terminates on the decimal place. QED
More generally, any integer, n, is equal to a product of powers of its prime divisors. Any terminating "decimal", base n, is equal to an integer times some power of n (the negative of the power of the last "decimal" place) and so is equal to that integer divided by that power of n. While some factors in denominator and numerator may cancel, that won't introduce new factors in the denominator. The only possible factors of the denominator are factors of n and so the only possible prime factors of the denominator are the prime factors of n.