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Jan. 31, 2023•0 likes•2 views

Jan. 31, 2023•0 likes•2 views

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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) 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. .

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- 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.