MathDB

Let

P = \{p_1,p_2,\ldots, p_{10}\}

be a set of

10

different prime numbers and let

A

be the set of all the integers greater than

1

so that their prime decomposition only contains primes of

P

. The elements of

A

are colored in such a way that:
[*] each element of

P

has a different color, [*] if

m,n \in A

, then

mn

is the same color of

m

n

, [*] for any pair of different colors

\mathcal{R}

and

\mathcal{S}

, there are no

j,k,m,n\in A

(not necessarily distinct from one another), with

j,k

colored

\mathcal{R}

and

m,n

colored

\mathcal{S}

, so that

j

is a divisor of

m

and

n

is a divisor of

k

, simultaneously.
Prove that there exists a prime of

P

so that all its multiples in

A

are the same color.

1