We need at least 10 more requests to produce the answer.
0 / 10 have requested this problem solution
The more requests, the faster the answer.
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)] 27. (a) Let...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
(i) For every nonzero integers a, b, prove that gcd(a, b) = gcd(−a, b). (ii) Show that for every nonzero integers a, b, a, b are relatively prime if and only if a and −b are relatively prime.
Part 15A and 15B (15) Let n E Z+,and let d be a positive divisor of n. Theorem 23.7 tells us that Zn contains exactly one subgroup of order d, but not how many elements Z has of order d. We will determine that number in this exercise. (a) Determine the number of elements in Z12 of each order d. Fill in the table below to compare your answers to the number of integers between 1 and d that are...
16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b) R(c, d) if and only if a c and b 3 d for all (a, b), (c, d)E A. Prove that R is a partial ordering on A that is not a total ordering. 16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b)...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
(d)n- 1013 2. Let a, b, c, d be integers. Prove the statement or give a counterexample (a) If (ab) c, then a |c and alc. (b) If a l b and c|d, then ac bod (c) If aYb and alc, then aYbc. (d) If a31b4, then alb. (e) If ged(a, b) 1 and alc and b c, then (ab) c. Here a and b are relatively prime integers, also called coprime integers.] rherF and r is an integer with...