Combinatorics | Number Theory Answers

Find the last three digits of the number 3×7×11×· · ·×2003. [Hint: Chinese

remainder theorem.]

There are 21 pupils in a grade 7 class. The class teacher has to choose eight of the pupils for a group that will visit Germany in three months time. In how many different ways can the teacher select pupils for the group ?

There are 5 different roads from city A to city B and 3 different roads from city B to city C. In how many ways can someone go from city A to city C passing by city B?

Determine whether -104 is a quadratic residue or non residue of the

prime 997.

Determine those odd primes p for which -3p=1 and those for which -

3p=-1

.Solve𝑥 ≡ 7(𝑚𝑜𝑑 11) 𝑥 ≡ 6(𝑚𝑜𝑑 8) 𝑥 ≡ 10(𝑚𝑜𝑑 15) Also find the

smallest non-negative solutions

Solve the following congruences x≡1 mod3, x≡2mod4

x≡3mod5

Prove that n=2 mod4,and when n=p ,where p is a prime and p≡3 mod4

Prove that the set of arithmetic functions form an abelian group under the

mapping Dirichlet multiplication with f1≠0

If m,n=1,show that m,φn=1