{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

strayer_p25

# strayer_p25 - Section 1.4 The Euclidean Algorithm 25 of 803...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Section 1.4 The Euclidean Algorithm 25 of 803 and 154 above is not unique. For example, the reader may verify that 11 = 19803 ~ 99154 is another such expression. In fact. there are inﬁnitely many expressions of 11 as an integral linear combination of 803 and 154. We will have more to say on this issue in Chapter 6. A useful programming project related to Examples 14 and 15 appears as Student Project 1. Exercise Set 1.4 54. Use the Euclidean algorithm (Theorem 1.13) to ﬁnd the greatest common divisors below. Express each greatest common divisor as an integral linear combination of the original integers. (a) (37, 60) (b) (78, 708) (c) (441,1155) (d) (793,3172) (e) (2059, 2581) (f) (25174, 42722) 55. Prove that 7 has no expression as an integral linear combination of 18209 and 19043. 56. Find two rational numbers with denominators 11 and 13, respectively, and a sum of 1%. 57. Use Exercise 51 and the Euclidean algorithm to ﬁnd the greatest common divisors below. Express each greatest common divisor as an integral linear combination of the original integers. (a) (221, 247, 323) (b) (210, 294, 490, 735) 58. [The following exercise presents an algorithm for computing the greatest common divisor of two positive integers analogous to the Euclidean algorithm. This new algorithm is based on the absolute least remainder algorithm given in part (c) of Exercise 15.] Let a, b e Z with a 2 b > 0. By the absolute least remainder algorithm, there exist q,, r1 6 Z such that lbl lbl “ZbCIi+riv Z?<r1—E If r1 74 0, there exist (by the absolute least remainder algorithm) qz, r2 5 Z such that |r I |r | b2r1Q2+rzy _?1<r25?1 If r2 # 0, there exist (by the absolute least remainder algorithm) q3, r35Z such that _ g < lrzl r1 : r2q3 + r3, 2 r3 5 ? Continue this process. (a) Prove that r,, 2 0 for some n. If n > 1, prove that (a, b) = |r,,all. (1)) Use the new algorithm above to ﬁnd (204, 228) and (233, 377). ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern