hw4 - (a) Suppose ( a,b ) = 1. Find all integer solutions...

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(1) Prove the converse to Wilson’s theorem: If ( m - 1)! ≡ - 1( mod m ) then m is prime. (2) Without using the uniqueness of prime factorization theorem prove that if a | m , b | m and ( a,b ) = 1 then ab | m . (3) Use Euclidean algorithm to express (a) (66 , 56) as 66 x + 56 y with integer x,y . (b) (900 , 120) as 900 x + 120 y with integer x,y (4) Let a,b be integers. Suppose ax 0 + by 0 = ( a,b ) where x 0 ,y 0 are provided by the Eclidean algorithm.
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Unformatted text preview: (a) Suppose ( a,b ) = 1. Find all integer solutions of ax + by = ( a,b ) (b) Find all integer solutions of ax + by = ( a,b ) in general (i.e without assuming that ( a,b ) = 1). (c) Find all integer solutions of 16 x + 6 y = (16 , 6). (5) Find the Euler function of each of the following numbers 48, 51, 101. 1...
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