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C Roettger Spring 2013 Math 350 – Practice Exam 1 Show all work, justify your conclusions. Calculators allowed, but no notes or books. All the best! Problem 1 Convert 2013 to binary and 1010001012 to decimal notation. Problem 2 a) Find integers s, t such that 120s + 63t = gcd(120, 63) b) Find the inverse of 32 modulo 43. Problem 3 Find an integer x < 120 such that x≡4 (mod 8) x≡2 (mod 3) x≡1 (mod 5) Problem 4 How many solutions does the congruence 12x ≡ 20 have modulo 40? Problem 5 Why is 21000 − 1 not a prime? Find a short proof (NOT trial division). Problem 6 a) Explain why any number with base 16-representation n = (a1 a2 . . . ak )16 satisfies n ≡ a1 + a2 + · · · + ak (mod 15) b) I wrote a number in base 16, n = A 46201B16 and unfortunately smudged the second digit. But I know that n is a multiple of 15. What was the missing digit? Problem 7 Let fn be the n-th Fibonacci number (so f0 = f1 = 1, fn+2 = fn+1 + fn ). Prove by induction that gcd(fn+1 , fn ) = 1 for all n ≥ 1, and that the Euclidean Algorithm finds this gcd in exactly n − 1 steps.