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Definition 2024

Euler's totient function ‎(uncountable)

(number theory) The function that counts how many integers below a given integer are coprime to it.
Due to Euler's theorem, if f is a positive integer which is coprime to 10, then
$10^{\phi (f)}\equiv 1{\pmod {f}}$
where $\phi$ is Euler's totient function. Thus $f|(10^{\phi (f)}-1)$ , 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 $aNM+bM+c$ . 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 $\lceil \log _{10}M\rceil$ digits.)