MathDB

Color the positive integers by four colors

c_1,c_2,c_3,c_4

. (1)Prove that there exists a positive integer

n

and

i,j\in\{1,2,3,4\}

,such that among all the positive divisors of

n

, the number of divisors with color

c_i

is at least greater than the number of divisors with color

c_j

3

. (2)Prove that for any positive integer

A

,there exists a positive integer

n

and

i,j\in\{1,2,3,4\}

,such that among all the positive divisors of

n

, the number of divisors with color

c_i

is at least greater than the number of divisors with color

c_j

A

9