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quiz3_solutions

quiz3_solutions - 8 → 5 but 3 × 2 n = 5 3 mod 8 i.e the...

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University of California, Santa Barbara CMPSC 40 SUMMER 2009 QUIZ III N A M E : 1. Suppose b is a positive integer. What is a pseudoprime to the base b ? Definition 1, page 240 of the text. 2. Suppose we are doing arithmetic with octal (base 8) numbers. Is the following multiplication correct? Explain. 17333 × 23173 = 440652471 No. “Casting out 7’s”, (17333) 8 (21) 8 3 (23173) 8 (20) 8 2 (440652471) 8 (41) 8
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Unformatted text preview: 8 → 5 but 3 × 2 n = 5. 3. Calculate 45 401 mod 8, i.e. the remainder when 45 401 is divided by 8. ±irst way: 45 401 ≡ 5 401 (mod 8) ≡ 5 · (5 2 ) 200 (mod 8) ≡ 5 · 1 200 (mod 8) ≡ 5 (mod 8) Second way: GCD(45, 8 )= 1 and φ (8) = 4 (Euler function). Therefore 45 4 ≡ 1 (mod 8) and 45 401 ≡ 45 · (45 4 ) 100 (mod 8) ≡ 45 (mod 8) ≡ 5 (mod 8)...
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