3. Let n be a positive integer. (a) Give the definition of a unit modulo n. For a E Z, show that [an E Zn is a unit modulo n if and only if a and n are coprime. (b) Give the definition of a primitive root modulo n. (c) State (without proof ) Hensel's lemma and Fermat's little theorem. (d) For which positive integers m do the following congruences have a solution? (i) 12x20 + x2 + 4x + 1= 0 mod 6m, (ii) x 103 + 2= 0 mod 5m, (iii) 24 - 2.3 + 2x2 - 2x + 1= 0 mod 3m. (e) Let n = pil . . .pth , where pi, . ..,PK are distinct primes and a1, . . ., ak are positive integers. (i) Define Euler's 4 function and give (without proof) a formula for (n) in terms of the p; and the ai. (ii) For which integers m do there exist infinitely many n with m p(n)? Explain your