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FinalReviewProbs

# FinalReviewProbs - MATH 135 Final Exam Extra Review...

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MATH 135 Fall 2008 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 non-negative remainder when 8 66 is divided by 17 . 3. Prove that if p and q are distict prime numbers, a is an integer and p | a and q | a, then pq | a. 4. Prove that if a and b are non-zero integers, then gcd( a, b - a ) = gcd( a, b ) . 5. (a) State Fermat’s Little Theorem. (b) Find the smallest non-negative integer congruent to 123 74 (mod 19) . (c) Find the remainder when 123 74 is divided by 37 . 6. Prove that 13 | (12 n + 14 n ) for all odd positive integers n. 7. Find all integers which simultaneously satisfy the linear congruences in the following system: 22 x 2 (mod 30) 5 x 7 (mod 18) 8. Solve the congruence x 3 + x 2 42 (mod 72) . 9. Solve x 63 5 (mod 13) , for x Z . 10. 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

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FinalReviewProbs - MATH 135 Final Exam Extra Review...

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