MATH 417 Midterm: MATH417 Midterm 3 2017 Spring Solution

• School: University of Illinois
• Department: Mathematics
• Course: MATH 417
• Professor: All

Document Summary

Math 417 spring 2017 section b1. Solutions: (25 points) determine all the ideals of z that contain the ideal. 60 = 22 3 5, it follows that gcd(24, 36, 60) = 22 3 = 12. Now, every ideal in z is principal, and we have (cid:104)a(cid:105) (cid:104)b(cid:105) if and only if b|a. Hence, the ideals that contain i are the principal ideals (cid:104)b(cid:105) with b|12, namely: (cid:104)1(cid:105), (cid:104)2(cid:105), (cid:104)3(cid:105), (cid:104)4(cid:105), (cid:104)6(cid:105), (cid:104)12(cid:105). Math 417 3rd midterm: (25 points) determine integers x and y such that gcd(135, 1987) = x135 + y1987. First, using the division algo- rithm, we obtain: 4 = 4 1 + 0 so we have: gcd(1987, 135) = gcd(135, 97) = gcd(97, 38) = gcd(38, 21) = = gcd(21, 17) = gcd(17, 4) = gcd(4, 1) = 1. = 17 4 (21 1 17) = 5 17 4 21. = 5 (38 1 21) 4 21 = 5 38 9 21.