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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.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 37E
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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).
Transcribed Image Text: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.
Transcribed Image Text: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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