Division and Congruence The primary difference between modular and normal arithmetic is, per-haps unsurprisingly, with regard to division.Theorem 3.9. If ka = kb (mod kn) then a = b (mod n).The modulus is divided by k as well as the terms, so the meaning of = changes. In Exercise 3.1.12you will prove this theorem, and observe that, in general, we do not expect a = b (mod n). 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6).(a) Check this explicitly using Theorem 3.6.(b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)?(c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not?(d) Prove Theorem 3.9.

Answered: Image /qna-images/answer/7bcb3bd8-65fa-4725-84f7-0924f3f6ad78.jpg

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this explicitly using Theorem 3.6. (b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)? (c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not? (d) Prove Theorem 3.9.

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this explicitly using Theorem 3.6. (b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)? (c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not? (d) Prove Theorem 3.9.

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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Question

Division and Congruence The primary difference between modular and normal arithmetic is, per-
haps unsurprisingly, with regard to division.
Theorem 3.9. If ka = kb (mod kn) then a = b (mod n).
The modulus is divided by k as well as the terms, so the meaning of = changes. In Exercise 3.1.12
you will prove this theorem, and observe that, in general, we do not expect a = b (mod n).

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6).
(a) Check this explicitly using Theorem 3.6.
(b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)?
(c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not?
(d) Prove Theorem 3.9.

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