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test2-fall90 - (b(5 points Does the equation 216 x 168 y =...

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MAS 3300 - TEST 2 There are 6 questions. The total number of points is 125. 100% = 100 points so you do not have to do all parts of all questions. Throughout this test all roman letters a, b, c, d, e, f, k, l, x, y, .. denote integers. 1. (15 points) (a) Find the gcd(216 , 168) using the Euclidean algorithm. (b) Find the prime factorizations of 216 and 168. (c) Find the gcd(216 , 168) using the prime factorizations. 2. (15 points) Find x and y such that 216 x + 168 y = 24. You may find your results from 1 useful. 3. (15 points) Show that if a , b are positive integers and a 2 | b 2 then a | b . 4. (15 points) Let a , b Z , a and b both not zero, and let J a,b := { ka + lb : k, l Z } . It can be shown that J a,b = J e for some e Z , where J e := { ke : k Z } . Prove that e | a and e | b . You may NOT assume Theorem 3.9. 1
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5. (15 points) (a) (10 points) Let a , b Z , a and b both not zero. Show that the equation ax + by = f has a solution iff f = k gcd( a, b ) for some k . (i.e. f is a multiple of gcd( a, b )) HINT: You may assume Theorem 3.9 this time.
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Unformatted text preview: (b) (5 points) Does the equation 216 x + 168 y = 16 have a solution ? Give reasons. 6. (50 points) Determine whether the Following are TRUE or ±ALSE. Give reasons. ±or each part the correct answer is worth 3 points and the reason is worth 2 points. (i) IF a | ( b + c ) then a | b and a | c . (ii) IF a | ( b + c ) then a | b or a | c . (iii) IF a | ( b + c ) then a | ( b 2 + 2 bc + c 2 ). (iv) IF a | c and b | c then ( a + b ) | c . (v) IF a | b and a | c then a | ( b + c ). (vi) IF a | c and b | c then ab | c . (vii) IF a and b are two diferent primes and a | c and b | c then ab | c . (viii) IF a | b and b | c then a | c . (ix) IF a | b and c | b then a | c . (x) There are no prime numbers between 10! + 3 and 10! + 9. 2...
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