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Unformatted text preview: NUMBERS AND SETS EXAMPLES SHEET 3. W. T. G. 1. Solve (ie., find all solutions of) the equations (i) 7 x 77 (mod 40). (ii) 12 y 30 (mod 54). (iii) 3 z 2 (mod 17) and 4 z 3 (mod 19). 2. Without using a calculator, work out the value of 17 10 , 000 (mod 31). 3. Again without using a calculator, explain why 23 cannot divide 10 881- 1. 4. Let a 1 = 6 and for n > 1 let a n = 6 a n- 1 . What is a 2002 (mod 91)? 5. An RSA encryption scheme ( n,d ) has modulus n = 187 and coding exponent d = 7. Factorize n , and hence find a suitable decoding exponent e . If you have a calculator, check your answer by encoding the number 35 and then decoding the result. 6. Let p be a prime number and let 1 6 k < p . Prove that ( p k ) is a multiple of p . If you use any results from the course, make clear what they are and how you are using them. 7. Let P be a polynomial of the form P ( x ) = x d + a d- 1 x d- 1 + ... + a 1 x + a , where the coefficients a i are all integers. Suppose that r and s...
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