10.1. Introduction to Linear Congruences

Try these first

1. 1.

   What is the form of a linear congruence?

   Hint

   Look for the equation that involves a modulus.

2. 2.

   Can a linear congruence have infinitely many solutions?

   Hint

   Consider the adjustments involving k.

3. 3.

   What is a linear congruence?

   • A polynomial equation
   • An equation in modular arithmetic
   • A geometric representation

   Hint

   Focus on the definition involving moduli.

4. 4.

   True or False: The Chinese Remainder Theorem assures a unique solution for any system of linear congruences.

   • True
   • False

   Hint

   Consider the conditions under which CRT applies.

5. 5.

   Find all integer solutions for the linear congruence $8x \equiv 16 \mod{48}$ and explain your steps.

   Hint

   Consider how to simplify the equation.

6. 6.

   Prove that if $a$ is a solution of the system of congruences from the Chinese Remainder Theorem, then any number of the form $a + lm$ (where $m$ is the product of modulus) is also a solution.

   Hint

   Write out what happens for each modulus.