Fall2004 Review with solutions

Fall2004 Review with solutions - MATH 135 Fall 2004 Final...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 non-negative 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 non-zero integers, then GCD( a, b- a ) = GCD( a, b ) . 6. (a) State Fermats 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 . 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)....
View Full Document

Page1 / 4

Fall2004 Review with solutions - MATH 135 Fall 2004 Final...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online