135_4_-_Solutions

# 135_4_-_Solutions - Assignment 4 Solutions 1 Suppose that m...

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Assignment 4 – Solutions 1. Suppose that m,n P . Prove that if m 2 6 | n 2 , then m 6 | n . (Hint: consider the contrapositive.) Solution: We prove the contrapositive: “If m | n , then m 2 | n 2 .” Since m | n , then n = qm for some q Z . Thus, n 2 = ( qm ) 2 = q 2 m 2 . Since q 2 Z , then m 2 | n 2 by deﬁnition. Since the contrapositive is true, then the original statement is true. 2. Find integers s and t such that ( - 5083) s + (1656) t = gcd( - 5083 , 1656) . Solution: The Euclidean Algorithm gives 5083 = 3 · 1656 + 115 , 1656 = 14 · 115 + 46 , 115 = 2 · 46 + 23 , 46 = 2 · 23 + 0 so we have gcd( - 5083 , 1656) = 23. Back-Substitution then gives rise to the sequence 1 , - 2 , 29 , - 89 so we have 5083 · 29 - 1656 · 89 = 23 and can take s = - 29 and t = - 89. Alternatively, the Extended Euclidean Algorithm gives rise to the table 1 0 5083 0 1 1656 1 - 3 115 - 14 43 46 29 - 89 23 So we have 5083 · 29 - 1656 · 89 = 23 and can take s = - 29 and t = - 89. 3. Prove that for all integers

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135_4_-_Solutions - Assignment 4 Solutions 1 Suppose that m...

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