Abstract Algebra (MATS2103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2016
MATS 203: ABSTRACT ALGEBRA
TUTORIAL SERIES 2
1. List 4 integers which are distinct but congruent to
(i) 4 modulo 9; (ii) -2 modulo 7; (iii) 3 modulo 11; (iv) -5 modulo 8.
2. Prove that
a
=
b(mod
15) ⟺
a ≡ b(mod
3) and
a ≡ b(mod
5).
3. Find the remainder when
(i) 10
417
is divided by 7, (ii) 4
407
is divided by 11, (iii) 8
341
is divided by 5.
4. What is the unit digit of (i) 3
1029
, (ii) 7
2124
.
5. Prove that if
a ≡ b(mod n)
and
x
is any integer, then
a
+
x = b
+ x (mod
n)
and
ax ≡ bx(mod
n), ∀x
ℤ.
6. Prove that
a ≡ a(mod
n), ∀a
ℤ
7. Show that 5
132
≡
4
(mod
127).
8. Prove that 10
m
+ 3.4
m+2
+ 5 is a multiple of 9, ∀m
ℕ
9. Prove that if
n
is a positive odd integer, then
(n + 1)
k
= (n
+ l)(mod 2n), ∀k
ℕ*.
10. If my birthday is 312 days from today and if today is Monday May 2003, on which day of the week will my birth
day fall?
11. Solve each of the following linear congruences, giving your answer in the form
x
=
c(mod n)
where 0 ≤
c < n.
(i)
89x ≡
1
9(mod
5), (ii) 97x
≡
13
(mod
105), (iii) 33x ≡ 21
(mod
5), (iv) 44x
≡
28
(mod
56),
(v)
54
≡
12
(mod
42), (vi) 42x
≡
18(mod 5), (vii) 33x
≡
11
(mod
36), (viii) 28x
≡
13
(mod
24)
12. Use the Euclidean algorithm to find the g.c.d. of
(i) 2695 and 1547; then find integers
p
and
q
such that 2695p+1547q =
d
where
d =
(2695,1547). Hence or
otherwise, solve the linear congruence 54x
≡
5(mod 31) giving
y
our answer in the form
x ≡ c(mod
31). Where
0 ≤c≤31
(ii) Repeat
(a)
for pairs of numbers 2695 and 1551 and the linear congruence 106a; = 7
(mod
61). Find all integer
solutions of the equation 1073x + 814y. How many integer solutions are there?
13. Given that
ac ≡ bc(mod n),
where
n
does not divide c and
d =
(c, n), n = md, prove that
a ≡ b(mod m).
14. Let k be a positive odd integer. Prove that (i)
k
2
≡
l (mod 4), (ii)
k
2
≡
k(mod 2k)
END
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