Daniel J Smith Custom TMUA Set A Paper 2 · Question 15

A non-empty finite set $S$ of whole numbers is called linked if and only if

for every $a$ in $S$ , there exists $b$ in $S$ with $b\neq a$ such that $\gcd(a,b)>1$

where $\gcd(a,b)$ denotes the greatest common divisor (also known as the highest common factor) of integers $a$ and $b$

Which of the following is true if and only if $S$ is not linked?

AFor every

a

S

, there exists

b

S

with

b\neq a

such that

\gcd(a,b)=1

BFor every

a

S

, there exists a prime factor of

a

that divides every element of

S

CThere exists

a

S

such that for every

b

S

with

b\neq a

we have

\gcd(a,b)>1

DThere exists

a

S

such that for every

b

S

with

b\neq a

we have

\gcd(a,b)=1

EThere exists a prime

p

such that

p

divides exactly one element of

S

examday