13.5. Multiplication Modulo k

Try these first

1. 1.

   What is the identity element in multiplication modulo 7?

   Hint

   Think about what number keeps others unchanged when you multiply.

2. 2.

   List the co-prime integers with respect to 4.

   Hint

   Consider what numbers have no common factors with 4.

3. 3.

   What is a group?

   • A set with one operation
   • A collection of elements that satisfies certain properties
   • A number line

   Hint

   Recall the definition of a group we discussed.

4. 4.

   True or False: The identity element in multiplication modulo k is always 0.

   • True
   • False

   Hint

   Think about what multiplying any number by 0 results in.

5. 5.

   Prove that if a and b are in ℤ* for some k, then the inverse of their product in multiplication modulo k is the product of their inverses.

   Hint

   Use the definition of inverses in group theory.

6. 6.

   Given k as a prime number, explain how ℤ* under multiplication modulo k simplifies to the entire non-zero set of integers less than k.

   Hint

   Utilize the property of primes in relation to factors.