Math 55 Worksheet
Adapted from worksheets by Rob Bayer, Summer 2009.
Warmup
1. Solve the congruence 5
x
≡
16(
mod
21)
x
=

4
Linear Congruences and the CRT
1. Solve the system of congruences
x
≡
3 (
mod
4)
x
≡
2 (
mod
5)
x
≡
5 (
mod
7)
Solution:
x
≡
47
mod 140.
2. Solve the system of congruences
2
x
≡
1 (
mod
3)
3
x
≡
2 (
mod
5)
4
x
≡
1 (
mod
7)
Solution:
To start this one, notice what happens if
we multiply both sides of the first equation by 2!
Since 2
·
2
≡
1
mod 3, the first equation is the same
as
x
≡
2
mod 3.
Similarly, we take care of the other equations.
Here’s a website which talks about the
CRT and has a calculator to help you solve problems like this:
(And plenty of other good resources too!
http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/chinese3.html)
3.
F
Suppose we want to find all solutions to
x
2
≡
4 (
mod
35)
(a) Show that such solution must also be a solution to the system
(
x
2
≡
4 (
mod
5)
x
2
≡
4 (
mod
7)
and viceversa.
(b) Show that, based on the conditions in (a),
x
≡
2 or 3 (
mod
5) and that
x
≡
2 or 5 (
mod
7).
(c) Use the CRT on each of the possibilities from (b) to find all solutions to
x
2
≡
4 (
mod
35).
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 Summer '09
 RobBayer
 Math, Logic, Congruence, Prime number, Square number, Ih

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