Final_1997_Ans

Final_1997_Ans - MATH 55 Prof Demmel Demmels Practice Final...

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MATH 55 Prof. Demmel Demmel’s Practice Final Solutions by Don Barkauskas and Josh Levenberg Question 1. 1. (a) one-to-one, not onto (b) one-to-one, not onto 2. (a) k c,d ( x )= 3 q x - d c (b) Yes, it is in one-to-one correspondence with the set of possible pairs ( c, d ) which is Z \{ 0 Z , the product of two countable sets. Question 2. 1. f ( x )is O ( g ( x )) if C, k > 0 such that x k , | f ( x ) |≤ C | g ( x ) | 2. (a) k = 3 2 (b) k =1 (c) k =0 (d) k =6 (e) no such k exists (f) no such k exists Question 3. 1. Any multiple of gcd(135 , 296) can be represented. 296 = 2 · 135 + 26 135 = 5 · 26 + 5 26 = 5 · 5+1 So gcd(135 , 296) = 1, and all integers can be so represented. 2. Solving the above for 1: 1=2 6 - 5 · 5 =2 6 - 5(135 - 5 · 26) 6 · 26 - 5 · 135 = 26(296 - 2 · 135) - 5 · 135 6 · 296 - 57 · 135 Now multiplying by 4 gives s = - 4 · 57 = - 228 and t =4 · 26 = 104 . 3. Directly applying the result from 3.2, x ≡- 228 (mod 296) Question 4. 1. 12 4 1( m o d 5 ) b yF L T , s o 1 2 198 12 4 · 49+2 (12 4 ) 48 · 12 2 2 2 4( m o d 5 ) , s ow e g e t 12 198 =4mod5 .
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This note was uploaded on 03/08/2010 for the course MATH 55 taught by Professor Strain during the Spring '08 term at Berkeley.

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Final_1997_Ans - MATH 55 Prof Demmel Demmels Practice Final...

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