Question

1.Let n be the decimal form of a rational number a/b, where a and b are nonzero integers.

a. If n is a terminating decimal, what can be said about the factors of b? Explain.
b. If n is a repeating decimal, what can be said about the number of digits in the repeating block? Explain.

Answer 1

A. Essentially, b is (or can be expressed in an equivalent fraction as) a power of ten. Say the decimal is 0.342, then the fraction would be 342/1000. Ten has two prime factors, 2 and 5. Some factors in increased powers of ten might be canceled out by the numerator, but factors will never be added. Thus, b must have prime factors of only 2 and 5 to terminate.
B. If n is a repeating decimal, the number of digits in the repeating block cannot be more than (b-1). For example, (1/7) has 6 digits in the repeating block.