Repeating Decimal - Reciprocals of Integers Not Co-prime To 10

Reciprocals of Integers Not Co-prime To 10

An integer that is not co-prime to 10 but has a prime factor other than 2 or 5 has a reciprocal that is eventually periodic, but with a non-repeating sequence of digits that precede the repeating part. The reciprocal can be expressed as:

where a and b are not both zero.

This fraction can also be expressed as:

if a > b, or as

if b > a, or as

if a = b.

The decimal has:

• An initial transient of max(a, b) digits after the decimal point. Some or all of the digits in the transient can be zeros.
• A subsequent repetend which is the same as that for the fraction .

For example 1/28 = 0.03571428571428…:

• the initial non-repeating digits are 03; and
• the subsequent repeating digits are 571428.