Think about it

Let the number I am thinking of be $x$ .

$xn+7=38$

$xn=31$

$31$ is prime, and since $x$ and $n$ are both integers, only the following combinations are possible for $\left[ x,n \right]$ : $\left[ 1,31 \right]$ $\left[ 31,1 \right]$ $\left[ -1,-31 \right]$ $\left[ -31,-1 \right]$

Therefore there are $\large \color{#20A900}{\boxed{ 4}}$ possible numbers I could be thinking of. You've just got to be careful about remembering the negative combinations!

Lets name the number he's thinking p pn+7=38 pn=31

Lets give 31 some values. Since 31 is a prime number, the only numbers that can be factors of it are 1 31 and -1 -31

Therefore he could be thinking just on those 4 numbers.

Answer is 4 only! Let my number be m then mn+7=38 then mn=31 so possible values of m are 1,-1,31 and -31 when n=31,-31,1 and -1 respectively!