Surname
Forename
ANSWER ALL QUESTIONS ON THIS QUESTION PAPER
NUMBER THEORY TEST
This test is worth 10% of the module.
1.
(a)
Let a, b, m and n with n  1 be integers. Prove the following congruent
results:
(i) If a  b  mod n  and m n then a  b  mod m .
(ii) If a  b  mod n  and c  0 then ca  cb  mod cn  .
[4 Marks]
[2 Marks]
(b)
Find all the solutions of the linear congruence
10 x  5  mod 35
Page 1 of 4
[6 Marks]
(c)
Find all the solutions of the linear congruence
5x  3  mod 21
[4 Marks]
2.
(a)
Fermat’s Little Theorem (FLT) is the following:
Let p be prime and suppose p does not divide into a. Then
a p1  1  mod p 
By using this (FLT) or otherwise determine the remainder when
1939170  2 is divided by 13.
Page 2 of 4
(b)
[7 Marks]
(i) Define Euler’s Phi function   n  where n
(ii) Show that if p is prime and n
then   p
n
 p
[2 Marks]
n
p
n 1
[4 Marks]
Page 3 of 4
(iii) Hence, or otherwise, determine  81 .
[2 Marks]
End of Test
Page 4 of 4