Fractions With Prime Denominators
A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The period of the repeating decimal of 1/p is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the period is equal to p − 1; if not, the period is a factor of p − 1. This result can be deduced from Fermat's little theorem, which states that 10p−1 = 1 (mod p).
The base-10 repetend (the repeating decimal part) of the reciprocal of any prime number greater than 5 is divisible by 9.
Read more about this topic: Repeating Decimal
Famous quotes containing the word prime:
“One’s prime is elusive. You little girls, when you grow up, must be on the alert to recognize your prime at whatever time of your life it may occur. You must then live it to the full.”
—Muriel Spark (b. 1918)