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quiz 3 solution

# quiz 3 solution - 22M:150:001 Quiz 3 Duration of quiz 10...

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22M:150:001 Quiz 3 Duration of quiz: 10 minutes. 19 th September 2012 Calculators, books or formula sheets are NOT allowed. Name: 1. Find the greatest common divisor of 2987 and 4669. (5 points. ) We’ll use the Euclidean Algorithm to determine the greatest common divisor. 4669 = 2987 · 1 + 1682 2987 = 1682 · 1 + 1305 1682 = 1305 · 1 + 377 1305 = 377 · 3 + 174 377 = 174 · 2 + 29 174 = 29 · 6 + 0 . Hence, the greatest common divisor (which by the Euclidean algorithm, is the last non- zero divisor) is 29. 1

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2. Show that log 5 7 is an irrational number. (5 points. ) We’ll prove this by contradiction. Let’s suppose that log 5 7 is not an irrational number, i.e., it is a rational number. Further, we note that since log
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Unformatted text preview: 5 7 > 0, we can express log 5 7 as a fraction p q with p,q ∈ N . log 5 7 = p q ⇒ 5 p q = 7 ⇒ 5 p = 7 q . Since 5 divides the left-hand-side, 5 must also divide the right-hand-side. (Notice that we are using the fact that p ∈ N to conclude 5 | 5 p and this is not true for p ≤ 0.) Therefore, we have that 5 | 7 q ⇒ 5 | 7 ··· 7 | {z } q times ⇒ 5 | 7 , (By Chap . 4 . 3 , Lemma 3) which is a contradiction as 7 is a prime. Therefore, our initial hypothesis must have been wrong, and hence log 5 7 is irrational. 2...
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• Fall '12
• Gupta
• Math, Prime number, Greatest common divisor, Continued fraction

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