1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots 1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n 2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
Advanced Calculus (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion. (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
Prove the following theorem using induction THEOREM 39. If a 70 and m, n e Z, then aman = am+n and (a")" = amn. Moreover, if a, n EN, then a" EN.
5. Prove that for n e Z, n is even, if and only if n2 is even. 6. Verify by induction that 3" > 2n? n>0.
10. Let a, b,n E Z such that n >0, n does not divide a and al B in Z/nZ. Assume a-and [N]-[a]. Prove n #313 and n 497, 4
1. Let n,m e N with n > 0. Prove that there exist unique non-negative integers a, ..., an with a: < 0+1 for all 1 Si<n such that m- Hint:(Show existence and uniqueness of a s.t. () <m<("), and use induction)