exam1h - MATH 470 Exam 1 Sections 501 200 You may use a...

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MATH 470 Exam 1: Sections 501, 200 You may use a calculator provided it only has the ability to add, multiply, subtract and divide. Some facts you might find useful [This test features the primes 17 , 211 and 479 ] ( mod 211) congruences 71457 139 , 17 8 170 , 17 53 48 , 17 17 92 53 44 170 , 44 53 172 , 172 53 52 , 53 139 56 , 139 139 9 ( mod 479) congruences 59757 361 , 59757 2 33 , 33 50 20 , 361 108 460 , 108 361 366 , 51 4 94647 , 10 8 42759 , 94647 . 42759 6175 139 211 319 , 17 211 241 , 139 17 294 , 17 17 74 ( mod 101069) congruences 51 2 2601 , 51 4 55841 , 10 6 90590 , 10 8 35567 , 55841 . 35567 36348 , 2601 . 90590 87400 , 2601 . 35567 22770 172 . 479 . 37 16286 , 84 . 211 . 460 67520 , 172 . 479 . 84 47900 , 172 . 211 . 84 16458 , 172 . 211 . 37 28907 . Q1: Q2: Q3: Q4: Q5: Q6: Q7: Q8: Q9: Q10: Total:
[1] Find d , the gcd of 3213 and 2720 and hence find integers x and y such that 3213 x +2720 y = d . [Show all steps of your computation]
[2] The Extended Euclidean algorithm is given the numbers 479 and 211 as input. It outputs the gcd and provides s and t such that 211 s +479 t = d : the gcd is 1 and s =84 , t = 37 . Use this to solve the congruence: 211 x 3( mod 479) . (5 P OINTS )
[3] Alice is using an affine cipher and a 27 character alphabet that sets the space =0 and a =1 , b = 2 , . . . , z =26 to send a message to Bob. You intercept the ciphertext QAQGREEZGUDW LXGWYOTZ. The "QAQ" could very well be "bob". If you make this assumption can you recover the key?

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