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Find T(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers." • 2250 199 • 286936650 • 22!

Find T(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers." • 2250 199 • 286936650 • 22!

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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Question 11

11. Find 1(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers." • 2250 • 199 • 286936650 • 22! 12. Let p : 17 and d be a divisor of o(p). Determine (d) for each d. List each element having order d, for all divisors, d, of o(p). 13. Calculate all the primitive roots of 41 and 26. 14. Demonstrate that 21 has no primitive root. pk 15. Let r be a primitive root of n. If gcd(a, n) = 1, then the smallest positive integer k such that a = (mod n) is called the index of a relative to r, denoted by ind,a. The theory of indices can be used to solve congruences. Consider the properties of indices (p. 164) and example 8.4 (p. 164). Solve 8x = 11 (mod 13) using the fact that 6 is a primitive root of 13. Construct a table of indices for 6 modulo 13. 16. (7 points (bonus)) Solve the exercise that was previously skipped (since only 14 are required). Be sure to clearly mark the exercise you wish to count for extra credit. Here is an excellent online large number calculator that has a built-in modulo n feature: https://www.calculator.net/big-number-calculator.html
Transcribed Image Text:11. Find 1(n), o(n), X(n), µ(n), w(n), and ø(n) for the following integers." • 2250 • 199 • 286936650 • 22! 12. Let p : 17 and d be a divisor of o(p). Determine (d) for each d. List each element having order d, for all divisors, d, of o(p). 13. Calculate all the primitive roots of 41 and 26. 14. Demonstrate that 21 has no primitive root. pk 15. Let r be a primitive root of n. If gcd(a, n) = 1, then the smallest positive integer k such that a = (mod n) is called the index of a relative to r, denoted by ind,a. The theory of indices can be used to solve congruences. Consider the properties of indices (p. 164) and example 8.4 (p. 164). Solve 8x = 11 (mod 13) using the fact that 6 is a primitive root of 13. Construct a table of indices for 6 modulo 13. 16. (7 points (bonus)) Solve the exercise that was previously skipped (since only 14 are required). Be sure to clearly mark the exercise you wish to count for extra credit. Here is an excellent online large number calculator that has a built-in modulo n feature: https://www.calculator.net/big-number-calculator.html
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