Download Exercises for Lectures 19 and 20

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.]
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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.
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