Desimal Berulang untuk suatu Numerator

  • Moh. Affaf STKIP PGRI BANGKALAN
Keywords: Decimal Representation, Midy's Theorem, Repeating Decimal.

Abstract

Let 𝑛 denote a positive integer relatively prime to 10 and π‘š be natural
number less than 𝑛 and relatively prime to 𝑛. Let the period of π‘š
𝑛
be π‘Žπ‘
with 𝑏 > 1. Break the repeating block of π‘Žπ‘ digits up into 𝑏 sub blocks,
each of length π‘Ž, and let 𝐡(π‘›π‘š, π‘Ž, 𝑏) denote the sum of these 𝑏 blocks.
In 1836 has been proved and re-proofed in 2013 that if 𝑝 is a prime
greater than 5, and the period of 1
𝑝
is 2π‘Ž, then 𝐡(𝑝1, π‘Ž, 2) = 10π‘Ž βˆ’ 1. In
2004, has been showed that if 𝑝 is a prime greater than 5, and the period
of 1
𝑝
is 3π‘Ž, then 𝐡(𝑝1, π‘Ž, 3) = 10π‘Ž βˆ’ 1. In 2005, also was showed that if
𝑝 is a prime greater than 5, and the period of 1
𝑝
is π‘Žπ‘, then 𝐡(𝑝1, π‘Ž, 3) is
multiply of 10π‘Ž βˆ’ 1. In 2007, study of repeating decimal of 1
𝑛
is
expanded, that 𝑛 don't have to prime greater than 5, but just relatively
prime to 10. In this paper, will be investigated property of repeating
decimal of π‘š
𝑛
with π‘š don't have to equal to 1, but enough less than 𝑛
and relatively prime to 𝑛.

Published
2018-07-31