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). ...
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 Spring '10
 BOYLAND

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