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Unformatted text preview: number when it is written in decimal form. Justify your answer. 9. Find all solutions to m 2 − n 2 = 88, for which both m and n are positive integers. 10. Prove that if m, n, d ∈ Z + and m ≡ n (mod d ), then gcd( m, d ) = gcd( n, d ) . 11. Use the Euclidean algorithm to determine whether or not the numbers 1947 and 2009 are relatively prime. 1 12. Use mathematical induction to prove that for all n ∈ Z + we have 21  (4 n +1 + 5 2 n1 ) . 13. For n ≥ 1, the n th harmonic number is de±ned as H n = 1 + 1 2 + 1 3 + ··· + 1 n . Use mathematical induction to prove that for all n ≥ 1 we have H 1 + H 2 + H 3 + ··· + H n = ( n + 1) H n − n . 14. The Fibonacci numbers are de±ned as: f 1 = 1 , f 2 = 1 , and f n = f n1 + f n2 , for n ≥ 3 . Use a proof by induction to show that the Fibonacci numbers satisfy 3  f 4 n , for all n ≥ 1 . 2...
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 Spring '09
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 Software engineering, Natural number, Euclidean algorithm, CONCORDIA UNIVERSITY DEPARTMENT, SOFTWARE ENGINEERING COMP232

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