| Line 8: | Line 8: | ||
Test using 3 as the generator: | Test using 3 as the generator: | ||
| − | 3^1 = 3 | + | 3^1 = 3; |
| − | 3^2 = 9 | + | 3^2 = 9; |
| − | 3^3 = 27 = -1 = 13 | + | 3^3 = 27 = -1 = 13; |
| − | 3^4 = 13*3 = 39 = 11 | + | 3^4 = 13*3 = 39 = 11; |
| − | 3^5 = 11*3 = 33 = 5 | + | 3^5 = 11*3 = 33 = 5; |
3^6 = 5*3 = 15 = 1 (end of cycle and 3 generates all values in U(14), therefore is a generator) | 3^6 = 5*3 = 15 = 1 (end of cycle and 3 generates all values in U(14), therefore is a generator) | ||
Test using 5 as the generator: | Test using 5 as the generator: | ||
| − | 5^1 = 5 | + | 5^1 = 5; |
| − | 5^2 = 25 = -3 = 11 | + | 5^2 = 25 = -3 = 11; |
| − | 5^3 = -15 = -1 = 13 | + | 5^3 = -15 = -1 = 13; |
| − | 5^4 = -1*5 = -5 = 9 | + | 5^4 = -1*5 = -5 = 9; |
| − | 5^5 = -5*5 = -25 = 3 | + | 5^5 = -5*5 = -25 = 3; |
5^6 = 3*5 = 15 = 1 (end of cycle and 5 generates all values in U(14), therefore is a generator) | 5^6 = 3*5 = 15 = 1 (end of cycle and 5 generates all values in U(14), therefore is a generator) | ||
To see if U(14) = <11> test the powers of 11 to see if they generate all values in U(14): | To see if U(14) = <11> test the powers of 11 to see if they generate all values in U(14): | ||
| − | 11^1 = 11 | + | 11^1 = 11; |
| − | 11^2 = -3*-3 = 9 | + | 11^2 = -3*-3 = 9; |
11^3 = 9*-3 = -27 = 1 (because <11> did not generate all values of U(14), <11> is not a generator) | 11^3 = 9*-3 = -27 = 1 (because <11> did not generate all values of U(14), <11> is not a generator) | ||
-K. Brumbaugh, 23:01, 3 February 2009 | -K. Brumbaugh, 23:01, 3 February 2009 | ||
Revision as of 19:03, 3 February 2009
Problem Statement: Show that U(14) = <3> = <5>. Is U(14) = <11>?
Answer: The long way of doing it (versus the tricks taught on 2/3/09):
U(14) = {1,3,5,9,11,13}
Test using 3 as the generator: 3^1 = 3; 3^2 = 9; 3^3 = 27 = -1 = 13; 3^4 = 13*3 = 39 = 11; 3^5 = 11*3 = 33 = 5; 3^6 = 5*3 = 15 = 1 (end of cycle and 3 generates all values in U(14), therefore is a generator)
Test using 5 as the generator: 5^1 = 5; 5^2 = 25 = -3 = 11; 5^3 = -15 = -1 = 13; 5^4 = -1*5 = -5 = 9; 5^5 = -5*5 = -25 = 3; 5^6 = 3*5 = 15 = 1 (end of cycle and 5 generates all values in U(14), therefore is a generator)
To see if U(14) = <11> test the powers of 11 to see if they generate all values in U(14): 11^1 = 11; 11^2 = -3*-3 = 9; 11^3 = 9*-3 = -27 = 1 (because <11> did not generate all values of U(14), <11> is not a generator)
-K. Brumbaugh, 23:01, 3 February 2009
