Exercises for Lectures 17 and 18 Number Theory II - Zn and Fermat’s Little Theorem 1. a) Use the Euclidean Algorithm to compute gcd(17, 23). b) Use your computation in part a) to find numbers λ and µ such that λ · 17 + µ · 23 = 1 c) Find the multiplicative inverse of 17 in Z23 . d) Find the multiplicative inverse of 23 in Z17 . [Note 23 ≡ 6 mod 17.] 2. a) Use the Euclidean Algorithm to compute gcd(41, 43). b) Use your computation in part a) to find numbers λ and µ such that λ · 41 + µ · 43 = 1 c) Find the multiplicative inverse of 41 in Z43 . d) Find the multiplicative inverse of 43 in Z41 . [Note 43 ≡ 2 mod 41.] 3. Find the multiplicative inverses of all non-zero elements of Z13 . You need not use the Euclidean Algorithm. 4. In Z24 find all elements which have a multiplicative inverse and, for each of these, give the inverse. 5. Find the solution to 3 · x ≡ 50 mod 101. Hint: Compute first the multiplicative inverse of 3 in Z101 . 6. Find the solution to 20 · x ≡ 81 mod 101. Find the solution to 20 · x ≡ 80 mod 101. Find the solution to 20 · x ≡ 5 mod 101. [Are any of your answers surprising?] 7. In class we considered the diagram obtained by multiplying elements of Z13 by 5, so each arrow represented multiplication by 5, e.g. 7 −→ 9 since 7 · 5 = 35 ≡ 9 mod 13. Draw the cycles for multiplication by 2? Draw the cycles for multiplication by 4? [Hint: The cycles for 2 should make the work much faster.] 8. Draw the multiplication diagram for the elements of Z12 in which the arrows denote multiplication by 2. If you multiply a ∈ Z12 by a power of 2 with exponent at least 2, then the result can only be one of three different elements of Z12 . What are those three elements? [Hint: The diagram should help.] 1 9. Let x ∈ Z101 , x 6= 0. Using Fermat’s Little Theorem, what can you say about x300 ? What can you say about x501 . 10. Let x ∈ Z103 , x 6= 0. Using Fermat’s Little Theorem, show that x101 is the multiplicative inverse of x. 2