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# hw5 - a n = d n = 1 5 Explain why 6 x â‰ 21 2 has no...

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MAT 312/AMS 351 – Fall 2010 Homework 5 1. Explain in your own words why, if n is a prime, a linear congruence equation ax n b always has a solution (i.e that given any integers a and b , there exists an integer x such that ax b is a multiple of n ), and that any two solutions are the same modulo n . 2. Solve 3 x 19 16. Not by trial-and-error please. 3. Solve 5 x 14 12 (14 not prime, but (5,14) = 1 sufficient). 4. Review the procedure for the case (
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Unformatted text preview: a, n ) = d n = 1. 5. Explain why 6 x â‰¡ 21 2 has no solutions. 6. Show that 6 x â‰¡ 21 9 if and only if 2 x â‰¡ 7 3. 7. Solve 2 x â‰¡ 7 3. Let x be the unique solution. 8. Check that x , x 1 = x + 1 Â· 7 and x 2 = x + 2 Â· 7 are all diferent solutions of 6 x â‰¡ 21 9. Why does this not work for x + 3 Â· 7? 9. Â±ind the seven solutions of 14 x â‰¡ 35 21. 1...
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