MAT 312/AMS 351
Solutions to Final Exam.
1. (30 points) a) (10 points) Solve the system of congruences
x
= 10
mod 13
,
x
= 16
mod 19
.
Answer:
x
= 244
mod 247.
Solutions:
1) Remark that
x
=

3
mod 13
, x
=

3
mod 19
,
so
x
=

3
mod 13
·
19
,
x
= 244
mod 247
.
2) Since 13 and 19 are coprime, we can present 1 as their linear combi
nation using Euclidean algorithm:
19 = 13 + 6
,
13 = 2
·
6 + 1
,
6 = 19

13
,
1 = 13

2(19

13) = 3
·
13

2
·
19
.
Therefore
x
= 3
·
13
·
16

2
·
19
·
10 = 624

380 = 244
.
b) (10 points) Solve the equation
14
x
= 6
mod 21
.
Answer:
No solutions.
Solution:
We have
d
=
g.c.d
(14
,
21) = 7 and 6 is not divisible by
d
.
Therefore the equation has no solutions.
c) (10 points) Solve the equation
14
x
= 6
mod 31
.
.
Answer:
x
= 27
mod 31
.
1
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Solution:
Since 14 and 31 are coprime, we can present 1 as their linear
combination using Euclidean algorithm:
31 = 2
·
14 + 3
,
14 = 4
·
3 + 2
,
3 = 2 + 1
,
3 = 31

2
·
14
,
2 = 14

4(31

2
·
14) = 9
·
14

4
·
31
,
1 = (31

2
·
14)

(9
·
14

4
·
31) = 5
·
31

11
·
14
.
Therefore

11
·
14 = 1
mod 31
,
x
=

6
·
11 =

66 = 27
mod 31
.
2. (25 points) a)(15 points) Find all positive integers
n
such that
n
2
+ 3
n
+ 2
is a prime number.
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 Summer '08
 BESCHER
 Algebra, Congruence, Prime number, congruences

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