Find the Least Common Multiple

To be divisible by both 12 and 15, a number must be divisible by their LCM. The prime factors are $12 = 2^2 \times 3$ and $15 = 3 \times 5$. The LCM is the product of highest powers of all prime factors present.

$\text{LCM}(12, 15) = 2^2 \times 3^1 \times 5^1 = 60$

Verify Divisibility for Options

Check if each option is a multiple of 60: $180/60 = 3$, $360/60 = 6$, $540/60 = 9$, and $750/60 = 12.5$.

$\frac{750}{60} = 12.5$

Conclusion

Since $750$ is not a perfect multiple of $60$, it is not divisible by both $12$ and $15$.

$750 \neq 60 \times k, \quad k \in \mathbb{Z}$