TMUA Official 2017 Paper 2 · Question 17

A set $S$ of whole numbers is called stapled if and only if for every whole number $a$ which is in $S$ there exists a prime factor of $a$ which divides at least one other number in $S$ .

Let $T$ be a set of whole numbers. Which of the following is true if and only if $T$ is not stapled?

AFor every number

a

which is in

T

, there is no prime factor of

a

which divides every other number in

T

BFor every number

a

which is in

T

, there is no prime factor of

a

which divides at least one other number in

T

CFor every number

a

which is in

T

, there is a prime factor of

a

which does not divide any other number in

T

DFor every number

a

which is in

T

, there is a prime factor of

a

which does not divide at least one other number in

T

EThere exists a number

a

which is in

T

such that there is no prime factor of

a

which divides every other number in

T

FThere exists a number

a

which is in

T

such that there is no prime factor of

a

which divides at least one other number in

T

GThere exists a number

a

which is in

T

such that there is a prime factor of

a

which does not divide any other number in

T

HThere exists a number

a

which is in

T

such that there is a prime factor of

a

which does not divide at least one other number in

T

examday

Official 2017 Paper 2 · Q17