1. Directions: Your solutions should be written out fully in paragraph form using complete
sentences and correct punctuation
Let n, p be elements of Z with n,p > 0 and p is prime.
(a) Prove or disprove: if a, b, c are elements of Z subscript n, with a not equal to 0 and ab=ac, the
b=c
(b) Prove or disprove: if a, b, c are elements of Z subscript p, with a not equal to 0 and ab=ac, the
b=c
For the disproof I just need an example with numbers that makes it not work
Solution
For a) the critical thing is for n not to be prime.
Consider n = 6
Let a = 2
Let b = 1
Let c = 4
Then, a * b = 2 in Z 6
However, a * c = 2 * 4 = 8 = 2 in Z 6
b) If a is not equal to 0. We shall prove the contrapositive, which proves the original statement.
Let ab = ac
Then a(b-c) = 0
Thus, p |a(b-c)

2. p does not divide a.
There is a unique prime factorization of a(b-c)
With this factorization, we add powers of the primes in a and the primes in b-c to get the prime
factorization.
Since p does not divide a, the power of p in the prime factorization of a must be 0.
Yet, as p divides a(b-c), the power of p in the prime factorization of a(b-c) must be greater than
or equal to 1. Thus, as 0 + the power of p in the factorization of b-c equals the power of p in the
prime factorization of a(b-c) > 0, the power of p in the factorization of b-c > 0.
If p divides b-c and 0 <= |b - c| <= p-1, b-c = 0. Thus, b = c.