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Find the following using a technique discussed in Section 8.4. 162 mod 45 = 164 mod 45 = 168 mod 45 = 1616 mod 45% = Need Help? Watch It Read It

Find the following using a technique discussed in Section 8.4. 162 mod 45 = 164 mod 45 = 168 mod 45 = 1616 mod 45% = Need Help? Watch It Read It

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the following using a technique discussed in Section 8.4. 162 mod 45 = 164 mod 45 = 168 mod 45 = 1616 mod 45 = Need Help? Watch It Read It
Transcribed Image Text:Find the following using a technique discussed in Section 8.4. 162 mod 45 = 164 mod 45 = 168 mod 45 = 1616 mod 45 = Need Help? Watch It Read It
Example 8.4.4 Computing a mod n When k Is a Power of 2 Find 144 mod 713. Solution Use property (8.4.1) to write 144 = (1442)?. Then 1444 mod 713 = (1442)² mod 713 = (1442 mod 713)? mod 713 = (20736 mod 713) mod 713 because 1442 = 20736 = 592 mod 713 because 20736 mod 713 = 59 = 3481 mod 713 because 592 = 3481 = 629 because 3481 mod 713 = 629. Example 8.4.5 Computing a" mod n When k Is Not a Power of 2 Find 123 mod 713. anon 2 Jeds Solution First write the exponent as a sum of powers of 2: 43 = 2° + 23 + 2+1 = 32+8+2+1. Next compute 12 for k = 0, 1, 2, 3, 4, and 5. 12 mod 713 = 12 122 mod 713 = 144 12 mod 713 = 144?mod 713 = 59 by Example 8.4.4 128 mod 713 = 592 mod 713 = 629 by Example 8.4.4 1216 mod 713 = 6292 mod T713 = 639 by the method of Example 8.4.4 1232 mod 713 = 6392 mod 713 = 485 by the method of Example 8.4.4. By property (8.4.2), 1243 = 1232+8 +2+1 = 1232.128. 12²- 12'. Thus, by Corollary 8.4.4, 1243 mod 713 = {(1232 mod 713)·(12° mod 713)·(122 mod 713)·(12 mod 713)} mod 713. By substitution, 1245 mod 713 = (485·629·144·12) mod 713 = 527152320 mod 713 = 48. You should know how to do the computations in Example 8.4.5 by hand using only a simple electronic calculator, but if you are computing a lot of residues, especially ones involving large numbers, you may want to write a short computer or calculator program to do the computations for you. Extending the Euclidean Algorithm An extended version of the Euclidean algorithm can be used to find a concrete expression for the greatest common divisor of integers a and b.
Transcribed Image Text:Example 8.4.4 Computing a mod n When k Is a Power of 2 Find 144 mod 713. Solution Use property (8.4.1) to write 144 = (1442)?. Then 1444 mod 713 = (1442)² mod 713 = (1442 mod 713)? mod 713 = (20736 mod 713) mod 713 because 1442 = 20736 = 592 mod 713 because 20736 mod 713 = 59 = 3481 mod 713 because 592 = 3481 = 629 because 3481 mod 713 = 629. Example 8.4.5 Computing a" mod n When k Is Not a Power of 2 Find 123 mod 713. anon 2 Jeds Solution First write the exponent as a sum of powers of 2: 43 = 2° + 23 + 2+1 = 32+8+2+1. Next compute 12 for k = 0, 1, 2, 3, 4, and 5. 12 mod 713 = 12 122 mod 713 = 144 12 mod 713 = 144?mod 713 = 59 by Example 8.4.4 128 mod 713 = 592 mod 713 = 629 by Example 8.4.4 1216 mod 713 = 6292 mod T713 = 639 by the method of Example 8.4.4 1232 mod 713 = 6392 mod 713 = 485 by the method of Example 8.4.4. By property (8.4.2), 1243 = 1232+8 +2+1 = 1232.128. 12²- 12'. Thus, by Corollary 8.4.4, 1243 mod 713 = {(1232 mod 713)·(12° mod 713)·(122 mod 713)·(12 mod 713)} mod 713. By substitution, 1245 mod 713 = (485·629·144·12) mod 713 = 527152320 mod 713 = 48. You should know how to do the computations in Example 8.4.5 by hand using only a simple electronic calculator, but if you are computing a lot of residues, especially ones involving large numbers, you may want to write a short computer or calculator program to do the computations for you. Extending the Euclidean Algorithm An extended version of the Euclidean algorithm can be used to find a concrete expression for the greatest common divisor of integers a and b.
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