Abstract Algebra (MATS2103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2017
MATS 203: ABSTRACT ALGEBRA
TUTORIAL SERIES IV
1.
List 4 integers which are distinct but congruent to
(i) 4 modulo 9; (ii) -2 modulo 7; (iii)3 modulo II: (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
447
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) bx( mod
n). Vx E Z.
6.
Prove that a≡a(mod n), Va € Z.
7.
Show that 5
152
≡ 4(mod 127).
8.
Prove that 10
352
+ 3.4
m+2
+ 5 is a multiple of 9. Vm € N.
9.
Prove that if n is a positive odd integer, then (n + l)
k
= (n + l)(mod 2n). Vfc € 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 ≡ 19(mod 5), (ii)97x ≡13(mod 105), (iii)33x ≡ 21(mod 5), (iv) 44x ≡ 28(mod 56),
(v) 54x ≡ 12(mod 42), (vi)42x ≡ I8(mod 5), (vii)33x ≡11(mod 36), (viii)28x ≡ I3(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 your 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 1O6
X
= 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), whore n does not divide c and d = (c,n),n = md, prove that n ≡ b{mod m).
14.
Let k be a positive odd integer. Prove that (i) k
2
≡ 1(mod 4), (ii) k
2
= k(mod 2k)
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