Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...
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- Prove that ||A||ı = max1Take non-zero positive integers m and n. Show that Z/mZxZ/nZZ/mnZ if and only if m and n are relatively prime, that is, gcd(m, n) = 1.Let n > 1 be an integer. Prove that ℤ /n has exactly n elements. Hint 1: Use the following strategy: 1) Prove that ℤ /n = {[0], [1], · · · , [n − 1]}. (This shows that |ℤ /n| ≤ n.) 2) Prove that [i] 6= [j] whenever 0 ≤ i < j ≤ n − 1. (This shows that [0], · · · , [n − 1] are all distinct, hence |ℤ /n| ≥ n.)+ CpXp + min ! x1,...,Xp C1x1+ .. subject to + alpXpLet A = {x | x = 12k for somek E Z} Let B = x | x = 4k for some k e Z} { - Prove that A CB.Show that ... image attachedLet n be a positive integer and X = {1,. X even if Y is even and odd if | Y| is odd. i. How many subsets does X have? 2n+1}. We will call a subset Y of ii. Prove that X has the same number of subsets of size n as subsets of size n + 1. iii. State and prove a formula for the number of even subsets of X. Hint: Find a bijection between the collection of even subsets and the collection of odd subsets.Let x, y, p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that for any a E Z: p|a² - pla.)Let a E Z, a #0. Prove that for any positive integers m, n with nSEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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