Question
(a) Prove that H is a group under modular addition. (b) Prove R is an equivalence relation on Z12.
(c) Give a full description of the equivalence classes given by R.
2) Let p, q > 1 be consecutive odd numbers.
(a) Show(p+q)² =4pq+4.
(b) Show(p+q)² ≡4 modpq.
(c) Show that 4−1 exists in Zpq.
(d) Show that the set
{x ∈ Z × pq|x = x⁻¹}
has at least three distinct elements in it. Hint: 1 and pq − 1 are two of them.
3) Let a be a positive integer. Prove that the sum of a consecutive integers is divisible by a if and only if a is odd.
Solution Preview
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.