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alg2 - Solutions Algebra hw Week 2 by Bracket 0.6 For...

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Solutions, Algebra hw, Week 2 by Bracket 0.6. For driver’s license numbers issued in New York prior to September of 1992, the three digits preceding the last two of the number of a male with birth month m and birth date b are represented by 63 m + 2 b . For females the digits are 63 m + 2 b + 1. Determine the dates of birth and sex(es) corresponding to the numbers 248 and 601. 248 : 248 = 63 · 3 + 2 · 29 + 1 March 29, Female 601 : 601 = 63 · 9 + 2 · 17 + 0 September 17, Male 0.13. Let n and a be positive integers and let d = gcd( a, n ). Show that the equation ax mod n = 1 has a solution if and only if d = 1. (From Gallian) Suppose that there is an integer n such that ab mod n = 1. Then there is an integer q such that ab - nq = 1. Since d divides both a and n , d also divides 1. So, d = 1. On the other hand, if d = 1, then by Theorem 0.2, there are integers s and t such that as + nt = 1. Thus, modulo n , as = 1. 0.16. Use the Euclidean Algorithm to find gcd(34 , 126) and write it as a linear combination of 34 and 126.
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