Discrete Mathematics with Graph Theory (3rd Edition) 220

Discrete Mathematics with Graph Theory (3rd Edition) 220 -...

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218 Solutions to Exercises (completing the induction) if we can prove that at most k steps are required in the calculations which begin with second line. This will follow by the induction hypothesis if 2 k - 1 ::; l ~ J < 2k. Now n < 2k+l so l!!J < !! < 2k. Also l!!J > !! - 1 > 2 k - 1 - 1 and this implies l!!J > 2 k - 1 - , 2 -2 ' 2 2 - 2 - , as desired. (b) Apply the Russian peasant method to compute the product of n = n x 1. The numbers which appear on the right in the computation are 2, 4, 8, ... , namely, powers of 2. Eventually, the product, n =
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Unformatted text preview: n x 1, is obtained as the sum of certain of these powers. By part (a), the number of powers involved is at most k, where 2 k-1 ::; n &lt; 2k. Since k -1 ::; 10g2 n, we know k ::; 1 + 10g2 n. 7. [BB] v'3 ~ 1.7321 as shown. 1. 7 5 3. 0 0 0 0 1 1 27 2 0 0 1 8 9 343 1 1 0 0 1 0 2 9 3462 7 1 0 0 6 9 2 4 34640 1 7 6 0 0 0 346405 1 7 6 0 0 0 0 1 7 3 2 0 2 5 2 7 9 8 5 .JI2 ~ 3.4641 as shown. 3. 4 0 12. 0 0 0 0 3 9 64 3 0 0 2 5 6 686 4 4 0 0 4 1 1 6 6924 2 8 4 0 0 2 7 6 9 6 69281 7 0 4 0 0 6 9 2 8 1 692820 1 1 1 9 0 0 I 0 1 1 1 9 0 0...
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