Repeating Decimal - Reciprocals of Composite Integers Coprime To 10

Reciprocals of Composite Integers Coprime To 10

If p is a prime other than 2 or 5, the decimal representation of the fraction has a specific period e.g.:

1/49 = 0.0204081 6326530 6122448 9795918 3673469 3877551

The period of the repeating decimal must be a factor of λ(49) = 42, where λ(n) is known as the Carmichael function. This follows from Carmichael's theorem, which states that: if n is a positive integer then λ(n) is the smallest integer m such that

for every integer a that is coprime to n.

The period of the repeating decimal of is usually pT_p where T_p is the period of the repeating decimal of . There are three known primes for which this is not true, and for which the period of is the same as the period of because p2 divides 10p−1−1; they are 3, 487 and 56598313 (sequence A045616 in OEIS).

Similarly, the period of the repeating decimal of is usually pk−1T_p

If p and q are primes other than 2 or 5, the decimal representation of the fraction has a specific period. An example is 1/119:

119 = 7 × 17

λ(7 × 17) = LCM(λ(7), λ(17))

= LCM(6, 16)

= 48

where LCM denotes the least common multiple.

The period T of is a factor of λ(pq) and it happens to be 48 in this case:

1/119 = 0.00840336 13445378 15126050 42016806 72268907 56302521

The period T of the repeating decimal of is LCM(T_p, T_q) where T_p is the period of the repeating decimal of and T_q is the period of the repeating decimal of .

If p, q, r etc. are primes other than 2 or 5, and k, ℓ, m etc. are positive integers then is a repeating decimal with a period of where, etc. are respectively the periods of the repeating decimals etc. as defined above.