TMUA Official Specimen Paper 2 · Question 17

Let $S$ be a set of positive integers, for example $S$ could consist of 3, 4, and 8.

A positive integer $n$ is called an $S$ -number if and only if for every factor $m$ of $n$ with $m>1$ , the number $m$ is a multiple of some number in $S$ .

So in the above example, 9 is an $S$ -number; this is because the factors of 9 greater than 1 are 3 and 9, and each of these is a multiple of 3.

Positive integer $n$ is therefore not an $S$ -number if and only if

Afor every (positive) factor

m

n

with

m>1

, there is a number in

S

which is not a factor of

m

Bfor every (positive) factor

m

n

with

m>1

, there is no number in

S

which is a factor of

m

Cfor every (positive) factor

m

n

with

m>1

, every number in

S

is a factor of

m

Dfor some (positive) factor

m

n

with

m>1

, there is a number in

S

which is not a factor of

m

Efor some (positive) factor

m

n

with

m>1

, there is no number in

S

which is a factor of

m

Ffor some (positive) factor

m

n

with

m>1

, every number in

S

is a factor of

m

examday

Official Specimen Paper 2 · Q17