South African Mathematics Olympiad

$128$ has factors that are all powers of $2$, i.e. $2^0,2^1,\dots ,2^7$.

The largest power of $2$ that divides $120$ is $2^3$.

The powers of $2$ dividing $128$ and exceeding $8$ are $2^4,2^5,2^6$ and $2^7$.

So, there are $4$ positive factors of $128$ that are not factors of $120$.