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Unformatted text preview: Solutions of Homework #4 : Proof Techniques Q1. Show that 5 √ 5 is irrational. Answer Proof by contradiction: Assume that 5 √ 5 = p q is in its simplest form, i.e., both p and q do not have a common divisor and therefore the fraction p q cannot be simplified further . Thus, 5 = p 5 q 5 (1) → p 5 = 5 q 5 (2) → 5  p 5 (3) → 5  p. (4) Where 5  p means that p is divisible by 5.................... .. ..(I) From (I), p = 5 k for some integer k . Substituting in (2): (5 k ) 5 = 5 q 5 → 5 5 k 5 = 5 q 5 → q 5 = 5 4 k 5 → 5  q 5 → 5  q. Thus, q is also divisible by 5................... ... .. (II) From (I) and (II), the fraction p q is not in its simplest form for it can be simplified further by dividing both the numerator and the denominator by 5 which contradicts the original assumption. 1 Q.2 The harmonic number H n is defined as for n ≥ 1 H n = n summationdisplay k =1 1 k ....
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 Fall '08
 W.Szpankowski
 Mathematical Induction, Natural number, Divisor, induction step, BASIS CASE

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