Combinatorics | Number Theory Answers

Show that among any group of five (not necessarily consecutive) integers, there

are two with the same remainder when divided by 4

How many committees of three(3) can be formed from eight(8) people?

Suppose p=0.9 and n=4 ,C={0000,1010,1100}.Compute pC,v where

v=1100

Three planets P1, P2, P3 orbit a star in a distant galaxy. The perihelion of a

planet is the point in the orbit of a planet that is closest to the star. The orbital periods

of P1, P2, P3 are 3, 5 and 11 years respectively. The most recent perihelia of these

planets were in the years shown in the table below. What is the next year in which all

three planets achieve perihelion simultaneously?

Prove that 3n+4n+5n is divisible by 12 whenever n is an odd positive integer.

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