romeo - Suppose a | b and c | b a Prove that c | b and a |...

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Math 210 Section 011: Test #2 NAME: This test has 6 pages . The points per page are 4, 5, 5, 4, 5, 5. [2 points] Question 1a. Compute gcd(161 , 196). [2 points] Question 1b. Write gcd(161 , 196) as an integer linear combination of 161 and 196. 1
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[3 points] Question 2a. For each of the following relations, determine whether it’s a function with domain { 1 , 2 , 3 , 4 } . (i) n (1 , 1) , (2 , 1) , (3 , 2) , (4 , 2) , (3 , 4) o (ii) n (1 , 2) , (2 , 3) , (3 , 2) , (4 , 3) o (iii) n (1 , 4) , (2 , 3) , (3 , 2) , (4 , 2) o [2 points] Question 2b. Suppose A and B are nonempty sets. Can A × B ever be a function from A to B ? Explain. 2
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[5 points] Question 3. Let A = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } . Define f : A A as follows: f ( x ) = 3 x if 0 3 x < 7 3 x - 7 if 7 3 x < 14 3 x - 14 if 14 3 x < 21 Express the inverse of f as a set of ordered pairs. 3
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[2 points] Question 4a.
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Unformatted text preview: Suppose a | b and c | b a . Prove that c | b and a | b c . [2 points] Question 4b. If a and b are relatively prime integers, prove that gcd( a + b,a-b ) = 1 or 2. 4 [5 points] Question 5. Prove that if n is an odd integer, then n 2-1 must be divisible by 8. 5 [5 points] Question 6. Consider the function f from N \ { 1 } to N defined by f ( n ) = the largest prime divisor of n . (i) Find f ( n ) for each n = 2 , 3 ,..., 10. (ii) What is the range of f ? (iii) Is f one-to-one? (iv) Is f onto? (v) Why did we not express f as a function from N to N ? 6...
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romeo - Suppose a | b and c | b a Prove that c | b and a |...

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