Discrete Mathematics with Graph Theory (3rd Edition) 82

Discrete Mathematics with Graph Theory (3rd Edition) 82 -...

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80 Solutions to Exercises 26. [BB] As suggested by the hint, consider the set S of all positive linear combinations of a and b. Since S contains a = 1a + Ob if a > 0 and -a otherwise, S is not empty so, by the Well-Ordering Principle, S contains a smallest element g. Since 9 E S, we know that 9 = ma + nb for integers a and b; hence, we have only to prove that 9 is th~ greatest common divisor of a and b .. First we prove that 9 I a. Write a = qg -+ r with 0 :5 r < g .. Note that r isa linear combination of a and b since r = a - qg = a - q(ma + nb)= (1 - qa)m + (-qn)b. Since 9 is the smallest positive linear combination of a and b, we have r = 0, so 9 I a as'desired.Similarly, 9 I b. Finally, if c I a and c I b, then c I (ma + nb), so c I g, so c:5 g. 27. [BB] Since gcd(63, 273) = 21, lcm(63, 273) = 63~t3) = 819 by formula (2) of this section. Since gcd(56, 200) = 8, lcm(56, 200) = .56(~OO) = 1400 by formula (2) of this section . . . :~ . ,. 28.
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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