
Residues and Multiples
really having trouble with these questions, help would be GREATLY appreciated!
1. For each positive integer m define Zm = {0, 1, 2, . . . , m − 1}, the set of all residues modulo m, and define
C(m) = { k ∈ Zm  0 6= k ≡ a3 (mod m) for some a ∈ Z }
the set of mod m residues that are nonzero cubes.
(i) Compute C(m) for all prime values of m less than 15.
(ii) Using results from the lectures determine the number of elements of C(m) when m is a prime number, treating separately the cases m ≡ 1 (mod 3) and m 6≡ 1 (mod 3).
(iii) If m = pq where p and q are primes such that p ≡ q ≡ 1 (mod 3), how many elements will C(m) have? Illustrate your answer by finding C(91) = C(7×13). (Hint: Make use of the Chinese Remainder Theorem.)
2. (i) It is well known that an integer n is a multiple of 9 if and only if the sum of its decimal digits is a multiple of 9. Prove this result. [Hint: Consider n − s, where s is the sum of all digits of n.]
(ii) Let n = (102451543210325435245123325435243520002503423542) 6. (Thus n is expressed in the base 6.) Is n multiple of 5? [Hint: Use (i) generalized]