9.5. Multiplicative Inverse Modulo N

Try these first

1. 1.

   What is the definition of a modular multiplicative inverse?

   Hint

   Think of it as a number that undoes the multiplication.

2. 2.

   If a = 3 and N = 11, what is the multiplicative inverse of 'a'?

   Hint

   Use trial and error to find b.

3. 3.

   What is the condition for a modular inverse to exist?

   • GCD(a
   • N) > 1
   • GCD(a
   • N) = 0
   • GCD(a
   • N) = 1

   Hint

   Think about independent numbers.

4. 4.

   True or False: If 'a' has an inverse modulo 'N', then GCD(a, N) can be greater than 1.

   • True
   • False

   Hint

   Consider what it means to be co-prime.

5. 5.

   Given a = 27 and N = 40, determine the multiplicative inverse using the Extended Euclidean Algorithm. Show all steps.

   Hint

   Break down each step and watch for the coefficients you get!

6. 6.

   If a = 12 and N = 30, can you find a multiplicative inverse? Justify your answer based on GCD.

   Hint

   Consider why common factors prevent inverses!