The LCM of two multiples of $15$ is $1200$. If the first number is $75$, which of the following is the second number?

Given:
First number $N_1=75$
LCM = $1200$
Both numbers are multiples of $15$.

Let the second number be $N_2=15k$.
We know that:

$$\text{LCM}(75,15k)=1200$$

$$\text{LCM}(15\times 5,15\times k)=15\times \text{LCM}(5,k)=1200$$

$$\text{LCM}(5,k)=\frac{1200}{15}=80$$

Since $80=5\times 16$, the value of $k$ must be a multiple of $16$. Let's test $k=16$:

$$N_2=15\times 16=240$$

Let's check the options. Option A is $240$.
Verification: $\text{LCM}(75,240)=1200$. This matches the given condition.