Homework 5 due:
April, 11, 2013
Problems marked with ∗ will be graded. Please practice on the rest of them to the extent you find
useful.
1. ∗Find all units and their inverses in Z8 .
2. Find inverses of all elements in Z7 .
3. ∗Prove that if [a], [b] ∈ Zn are units, then [a] ⊙ [b] is a unit.
4. Let n ∈ Z, n ≥ 2, prove that for any [a], [b], [c] ∈ Zn
([a] ⊙ [b]) ⊙ [c] = [a] ⊙ ([b] ⊙ [c])
5. Using the recipe from the proof of Chinese remainder theorem, find [x] ∈ Z90 such that
1. [x] = [1] in Z2
2. [x] = [3] in Z5
3. [x] = [5] in Z9
6. ∗Is it possible to find [x] ∈ Z8 such that
1. [x] = [1] in Z2
2. [x] = [2] in Z4 ?
7. ∗Find a primary representative for [8100342 ] in Z11 . In other words find r ∈ {0, 1, 2, ..., 10} such that
[8100342 ] = [r] in Z11 . Hint: Use Fermat’s Little Theorem.
8. Consider Zp , p is a positive prime number. Using the fact that
[a] ⊙ [b] = [0] ⇒ [a] = 0 ∨ [b] = 0
prove by induction on n that
[a1 ] ⊙ [a2 ] ⊙ . . . ⊙ [an ] = [0] ⇒ [a1 ] = [0] ∨ [a2 ] = [0] ∨ . . . ∨ [an ] = [0]
9. ∗Using the fact that
gcd(m, a) = 1 ∧ gcd(m, b) = 1 ⇒ gcd(m, a ⋅ b) = 1
prove by induction on n that
gcd(m, a1 ) = 1 ∧ gcd(m, a2 ) = 1 ∧ . . . ∧ gcd(m, an ) = 1 ⇒ gcd(m, a1 ⋅ a2 ⋅ . . . ⋅ an ) = 1
10. ∗Let m ∈ Z. Prove that for [m] = [m2 ] in Z2 . This statement means that m is even iff m2 is even.
1