MATH135 Lecture Notes - Lecture 7: Multiplication Table, Composite Number
14 May 2018
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MATH 135 Fall 2015: Extra Practice Set 7
These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike
assignment problems, they may discussed in depth on Piazza.
•The warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any
of them, then you should review your notes and possibly speak to an instructor before beginning the
corresponding assignment.
•The recommended problems supplement the practice gained by doing the corresponding assignment.
Some should be done as the material is learned and the rest can be left for exam preparation.
•A few more challenging extra problems are also included for students wishing to push themselves
even harder. Do not worry if you cannot solve these more difficult problems.
Warm-up Exercises
1. For each linear congruence, determine the complete solution, if a solution exists.
(a) 3x≡11 (mod 18)
(b) 4x≡5 (mod 21)
2. Complete a multiplication table for Z5.
3. What is the remainder when 1443 is divided by 41?
Recommended Problems
1. How many integers xwhere 0 ≤x < 1000 satisfy 42x≡105 (mod 56)?
2. State whether the given statement is true or false and prove or disprove accordingly.
(a) For all a, b, c, x ∈Zsuch that c, x > 0, if a≡b(mod c) then a+x≡b+x(mod c+x).
(b) For all m∈Nand for all [a]∈Zmthere exists a [b]∈Zmsuch that [b]2= [a].
3. In each of the following cases, find all values of [x]∈Zm, 0 ≤x<m, that satisfy the equation.
(a) [4][3] + [5] = [x]∈Z10
(b) [7]−1−[2] = [x]∈Z10
(c) [2][x] = [4] ∈Z8
(d) [3][x] = [9] ∈Z11
4. Which elements of Z6have multiplicative inverses?
5. What are the integer solutions to x2≡1 (mod 15)?
6. What are the last two digits of 43201?
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