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final-sp02 - 1[20 points Let a b c Z with(a b = 1...

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1. [20 points] Let a , b , c Z with ( a, b ) = 1. Prove (without assuming the Fundamental Theorem of Arithmetic) that if a | c and b | c then ab | c . HINT : Since ( a, b ) = 1 there are integers x , y such that ax + by = 1. 2. [20 points] Let a , b , c Z and assume that a and b are not both zero and let d = ( a, b ). Prove that ax + by = c for some integers x and y if and only if d | c . HINT : Use the result that ( a, b ) = min { ma + nb : m, n Z , ma + nb > 0 } . 3. [5 + 15 = 20 points] (i) Define pseudoprime . (ii) Prove that 161038 = (2)(73)(1103) is a pseudoprime. 4. [5 + 15 = 20 points] (i) State Euler’s Theorem. (ii) Let m , n be positive relatively prime integers. Prove that m φ ( n ) + n φ ( m ) 1 (mod mn ) . 5. [4 × 5 = 20 points] (i) Define what it means for an arithmetic function to be multiplicative . (ii) Prove that if f is a multiplicative function and f (1) = 0 then f ( n ) = 0 for all n . (iii) Prove that if f is a multiplicative function and f (1) 6 = 0 then f (1) = 1. (iv) Suppose f ( n ) and g ( n ) are multiplicative functions. Prove that h ( n ) = f ( n ) g ( n ) is multiplicative. 6. [15 + 5 = 20 points] (i) Let m , n be positive integers with m | n . Prove that φ ( m ) | φ ( n ) . (ii) Prove or disprove the converse of (i). 1
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2 7. [20 points] Let n Z with n > 1. If f
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