test2-fall90 - (b) (5 points) Does the equation 216 x + 168...

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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 if 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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test2-fall90 - (b) (5 points) Does the equation 216 x + 168...

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