MathDB

Let

S \subset \{ 1, \dots, n \}

be a nonempty set, where

n

is a positive integer. We denote by

s

the greatest common divisor of the elements of the set

S

. We assume that

s \not= 1

and let

d

be its smallest divisor greater than

1

. Let

T \subset \{ 1, \dots, n \}

be a set such that

S \subset T

and

|T| \ge 1 + \left[ \frac{n}{d} \right]

. Prove that the greatest common divisor of the elements in

T

1

. ———-- [Second Version] Let

n(n \ge 1)

be a positive integer and

U = \{ 1, \dots, n \}

. Let

S

be a nonempty subset of

U

and let

d (d \not= 1)

be the smallest common divisor of all elements of the set

S

. Find the smallest positive integer

k

such that for any subset

T

U

, consisting of

k

elements, with

S \subset T

, the greatest common divisor of all elements of

T

is equal to

1

Problem Statement