13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1 and every prime number p, if po-pb modulo n, then a b m0dulo n 14. (i) Suppose that r0 and y2 0. What is the least natural number n such that (0? prove your claim (ii) Explain why for s values of r, one has (1+)"1 nr (ii) Use the estimation o) to estimate 1.002. What is the exact error involved in this estimate?
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1 and every prime number p, if po-pb modulo n, then a b m0dulo n 14. (i) Suppose that r0 and y2 0. What is the least natural number n such that (0? prove your claim (ii) Explain why for s values of r, one has (1+)"1 nr (ii) Use the estimation o) to estimate 1.002. What is the exact error involved in this estimate?