7.5. Algorithms for Modular Arithmetic

Try these first

1. 1.

   What is 7 modulo 5?

   Hint

   Think about how many times 5 fits into 7.

2. 2.

   What is the result of -10 modulo 4?

   Hint

   Visualize moving clockwise on a clock face analogy.

3. 3.

   What is 9 modulo 4?

   • 1
   • 2
   • 3

   Hint

   Consider how many times 4 goes into 9.

4. 4.

   True or False: Modular division is equivalent to normal division.

   • True
   • False

   Hint

   Remember that dividing can lead to fractions in modular arithmetic.

5. 5.

   Prove that if a ≡ b mod N then a - b is divisible by N. Provide a detailed explanation.

   Hint

   Consider the definitions of modulus and how remainders work.

6. 6.

   Using the square-and-multiply method, compute 3^23 mod 11 step by step.

   Hint

   Break down into binary steps and gather non-zero contributions.