# hw3 - (6 Use Euclidean algorithm to ﬁnd(66 56 Extra...

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(1) Show that if ( a,b,c ) is a Pythegorean triple such that a 2 + b 2 = c 2 then a and b can not both be odd. (2) Show that lcm ( a,b ) = ab ( a,b ) for any natural numbers a,b . Here lcm ( a,b ) is the least common multiple of a and b . (3) Find the rule for checking when an integer is divisible by 7 similar to the rule for checking divisibility by 11 done in class. (4) To what number between 0 and 6 inclusive is the product 11 · 18 · 2322 · 13 · 19 congruent modulo 7? (5) To what number between 0 and 4 inclusive is the sum 1 + 2 + 2 2 + 2 3 + ... + 2 219 congruent modulo 5?
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Unformatted text preview: (6) Use Euclidean algorithm to ﬁnd (66 , 56). Extra Credit (to be written up and submitted separately) : We extend the notion of prime and composite numbers to negative integers as follows: An integer n diﬀerent from 0 , ± 1 is prime (respectively composite) if | n | is prime (respectively composite). Let f ( x ) be a non-constant polynomial with integer coeﬃcients. Prove that there exists an integer x such that f ( x ) is composite. 1...
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