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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