MathDB

Let

p

and

q

be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets

A

B

C

such that for each integer

z

in each of the sets

A

B

C

there is exactly one of the numbers

z

z+p

and

z+q

. a) Prove that such a decomposition is possible if and only if

p+q

is divisible by

3

. b) In the case we omit the restriction that

p

q

may not have a common divisor, prove that for

p \neq q

the number

\frac{p+q}{\gcd(p,q)}

is divisible by 3.

Problem Statement