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1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication we mean +6 and 6).

1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication we mean +6 and 6).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication we mean +6 and 6). 2. Let f : Z? –→ Z² be given by f(m,n) = (m-n,n). The composite functions fk, for k e Z+, are defined as fi(m, n) = f(m,n), and fk+1(m,n) = f(fr(m, n)), for k e Z*. Give a formal proof by induction that fr(m,n) = (m – kn, n) , for all k e Z+. - 3. Use induction to show that for all positive integers n (a) 13 + 23 + 33 + ... + n³ = (n(n + 1)/2)². (b) 1·1! + 2 - 2! + ... + n· n! = (n + 1)! – 1 (c) if n > 6, then 3" < n!
Transcribed Image Text:1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication we mean +6 and 6). 2. Let f : Z? –→ Z² be given by f(m,n) = (m-n,n). The composite functions fk, for k e Z+, are defined as fi(m, n) = f(m,n), and fk+1(m,n) = f(fr(m, n)), for k e Z*. Give a formal proof by induction that fr(m,n) = (m – kn, n) , for all k e Z+. - 3. Use induction to show that for all positive integers n (a) 13 + 23 + 33 + ... + n³ = (n(n + 1)/2)². (b) 1·1! + 2 - 2! + ... + n· n! = (n + 1)! – 1 (c) if n > 6, then 3" < n!
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