The greatest number that divides $155$ and $307$ leaving remainders $5$ and $7$, respectively, is:

To find the required greatest number, we need to subtract the respective remainders from the given numbers and then calculate their Highest Common Factor ($\text{HCF}$):
$$155-5=150$$
$$307-7=300$$
Now, calculate the $\text{HCF}$ of $150$ and $300$:$150=150\times 1$$300=150\times 2$ Clearly, $\text{HCF}(150,300)=150$.