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The University of Sydney
MATH2068/2988 Number Theory and Cryptography
(http://www.maths.usyd.edu.au/u/UG/IM/MATH2068/)
Semester 2, 2014
Lecturer: A.Fish
Assignment 2
1.
Due: 24/10/2014
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]
3
(MATH2988) Let a be an integer greater than 1 and let p be an odd prime that
is not a divisor of a(a2 − 1). Let n = (a2p − 1)/(a2 − 1). Prove that an−1 ≡ 1
(mod n), by showing that 2p|(n − 1) and a2p ≡ 1 (mod n).