7.1. Discrete Mathematics

Try these first

1. 1.

   What is 7 mod 3?

   Hint

   Find the remainder when dividing 7 by 3.

2. 2.

   What is the congruence relation for 10 and 4 under modulo 6?

   Hint

   Check if their difference is divisible by 6.

3. 3.

   What is the result of 12 mod 5?

   • 1
   • 2
   • 3
   • 4

   Hint

   Perform the division and check the remainder.

4. 4.

   True or False: Division in modular arithmetic is always well-defined.

   • True
   • False

   Hint

   Consider divisibility and inversibility.

5. 5.

   Illustrate how the square-and-multiply method can change the efficiency of calculating 2^20 mod 7.

   Hint

   Follow the binary positions of 1s to determine which powers to use.

6. 6.

   Challenge yourself: Calculate and explain why 15 and 27 are not congruent modulo 6.

   Hint

   Check their remainders under the modulus.