Problem 18. If gcd(a, 35) = 1, show that a^12 ? 1 (mod 35). Hint: first show a^12 ? 1 (mod 5) a^12 ? 1 (mod 7).
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a) Prove the cancelation law for congruences: If ax ≡ ay (mod m) and gcd(a,m) = 1 then x ≡ y (mod m), by making use of the multiplicative inverse of a. b) Show that the cancelation law fails if gcd(a,m) > 1 by considering the example 6x ≡ 6y (mod 8). (That is, show there is a solution with x ≠ y (mod 8).)
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Using the method shown in class, which first uses the Euclidean algorithm and then goes back up the list of equations, express the gcd(35, 78) as a linear combination of 35 and 78. Show your work. Based on your answer to the previous problem, can you find an inverse of 35 modulo 78?
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