Homework 4 - Homework 4: Solutions Section 2.8: 1, 4, 9 1....

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Homework 4: Solutions Section 2.8: 1, 4, 9 1. Find a primitive root of the prime 3; the prime 5; the prime 7; the prime 11; the prime 13. 3: 2 (this is the only primitive root mod 3 as φ (3 - 1) = φ (2) = 1) 5: 2 and 3 are primitive roots (again, notice that φ (5 - 1) = φ (4) = 2, so these are the only ones) 7: 3 and 5 are primitive roots ( φ (7 - 1) = φ (6) = 2, thus there are no others) 11: Notice that 11 - 1 = 10 = 2 · 5. We know that if an element a has order h , and ( a, 11) = 1, then h | φ (11), i.e. h | 10. Since (2,11)=1, the order of 2 is 1, 2, 5, or 10. Since 2 1 2( mod 11), 2 2 4( mod 11), and 2 5 = 32 ≡ - 1( mod 11), it must be that 2 10 1( mod 11). All in all, 11 has 4 primitive roots: 2, 6, 7, 8 13: Notice 13 - 1 = 12 = 2 2 · 3, so there are 4 primitive roots. 2 6 - 1( mod 13), and 2 4 3( mod 13), 2 3 8( mod 13), 2 2 4( mod 13), so the order of 2 must be 12. The primitive roots of 13 are: 2, 6, 7, 11 ...

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This note was uploaded on 06/15/2009 for the course MATH 115 taught by Professor Mok during the Fall '07 term at University of California, Berkeley.

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Homework 4 - Homework 4: Solutions Section 2.8: 1, 4, 9 1....

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