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• Due to Euler's theorem, if f is a positive integer which is coprime to 10, then    10^{φ(f)} ≡ 1 \pmod f where φ is Euler's totient function. Thus f | (10^{φ(f)} - 1), which fact which may be used to prove that any rational number whose expression in decimal is not finite can be expressed as a repeating decimal. (To do this, start by splitting the denominator into two factors: one which factors out exclusively into twos and fives, and another one which is coprime to 10. Secondly, multiply both numerator and denominator by such a natural number as will turn the first said factor into a power of 10 (call it N). Thirdly, multiply both numerator and denominator by such a number as will turn the second said factor into a power of 10 minus one (call it M). Fourthly, resolve the numerator into a sum of the form a N M + b M + c. Then the repeating decimal has the form a.b(c) where b may be padded by zeroes (if necessary) to take up \log_{10} N digits, and c may be padded by zeroes (if necessary) to take up ⌈ \log_{10} M⌉ digits.)