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Unformatted text preview: MATH 135 Fall 2004 Final Exam Extra Review Problems 1. State, with reasons, the number of integral solutions to the equation 7! x + 4 3 y = 88 . 2. Find the least nonnegative remainder when 8 66 is divided by 17 . 3. (a) Add the following integers and express your answer as a base 2 integer: (4 F A ) 16 , (336) 7 , (1221) 3 . Recall that A = 10 , B = 11 , C = 12 , D = 13 , E = 14 , F = 15 . (b) Evaluate ([10] + [6] 1 ) ([9] [3][6]) in Z 11 . 4. Prove that if p and q are distict prime numbers, a is an integer and p  a and q  a, then pq  a. 5. Prove that if a and b are nonzero integers, then GCD( a, b a ) = GCD( a, b ) . 6. (a) State Fermat’s Little Theorem. (b) Find the smallest nonnegative integer congruent to 123 74 (mod 19) . (c) Find the remainder when 123 74 is divided by 37 . 7. Prove that 13  (12 n + 14 n ) for all odd positive integers n. 8. Find all integers which simultaneously satisfy the linear congruences in the following system: 22 x ≡ 2 (mod 30) 5 x ≡ 7 (mod 18) 9. Solve the congruence x 3 + x 2 ≡ 42 (mod 72) . 10. Solve x 63 ≡ 5 (mod 13) , for x ∈ Z . 11. Let c represent a fixed integer. Find all the integers x that satisfy the following system of linear congruences. x ≡ 1 (mod 8) x ≡ c + 1 (mod 11) Express your answer in the form: x is congruent to some expression involving c, modulo an appro priate modulus. 12. If a, b, c ∈ Z such that a  c and b  c and GCD( a, b ) = 1, then ab  c . 13. Find the complete solution of 7 x 2 ≡ 2 x (mod 101)....
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This note was uploaded on 10/21/2010 for the course MATH 135 taught by Professor Andrewchilds during the Fall '08 term at Waterloo.
 Fall '08
 ANDREWCHILDS
 Remainder, Integers

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