Congruences (1.7-1.14)

• Exercise 1.7. Apply the definition carefully.
  • Consider variations to the exercise: If $45\equiv 9 \pmod n$, then what values could $n$ possibly take?
  • A Special case: When is a number congruent to 0?
• Exercise 1.8. Again, apply the definition.
• Theorem 1.9.
• Theorem 1.10. Is the theorem also true as an “if and only if”? Why did the book not state it that way?
• Theorem 1.11. Pay attention to the note that follows it.
• Theorem 1.12.
  • Consider the special case of this theorem where $c=d$. State how that theorem would look like then prove it.
  • Can you use this special case to provide a proof of the general case?
• Theorem 1.13. Provide both a direct proof as well as a proof that uses theorem 1.12.
• Theorem 1.14. Here again consider the special case of the theorem, and how it may help prove the more general case.
• Consider the following question: If $ac \equiv bc \pmod n$, does it necessarily follow that $a\equiv b \pmod n$? How does this relate to previous work?
• Find ways to express in English all the above theorems, without having to resort to individual variable names.