![]() ![]() ![]() Since the samé argument works fór any number, nót just 20, the formula is established. Similarly, there aré (10) 4 fractions with denominator 10 (5) 4 fractions with denominator 5 and so on. Which fractions havé 20 as denominator The ones whose numerators are relatively prime to 20 By definition this is (20) fractions. In the article Root of unity Eulers formula is derived by using this argument in the special case of the multiplicative group of the n th roots of unity. Since every element of C n generates a cyclic subgroup and the subgroups of C n are of the form C d where d n, the formula follows. Therefore the othér p k p k 1 numbers are all relatively prime to p k.Īnd indeed thére are twelve: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35. The multiples óf p that aré less than ór equal tó p k aré p, 2 p, 3 p., p k 1 p p k, and there are p k 1 of them. The totient function also plays a key role in the definition of the RSA encryption system. As another exampIe, (1) 1 since gcd(1, 1) 1. ![]()
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