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Unformatted text preview: MATH 135 Fall 2007 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 FA ) 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). 14. A sequence of integers a 1 ,a 2 ,a 3 ,... is defined by a 1 = 0 ,a 2 = 1 , and a n +2 = 2 a n +1 + a n for n ≥ 1 . Prove that 5 2 n > a n for all n ≥ 1 ....
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This note was uploaded on 10/24/2009 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.
 Winter '08
 ANDREWCHILDS
 Remainder, Integers

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