Find the Least Common Multiple (LCM)

To find the smallest number divisible by $12$, $15$, and $18$, we first calculate their $\text{LCM}$.

• Prime factorization of $12 = 2^2 \times 3^1$
• Prime factorization of $15 = 3^1 \times 5^1$
• Prime factorization of $18 = 2^1 \times 3^2$ Taking the highest power of each prime factor gives the $\text{LCM}$.

$\text{LCM}(12, 15, 18) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180$

Convert the LCM into the Smallest Perfect Square (M)

For a number to be a perfect square, all the exponents in its prime factorization must be even numbers. In $180 = 2^2 \times 3^2 \times 5^1$, the factor $5$ has an odd exponent ($1$). To make it a perfect square, we must multiply $180$ by $5$.

$M = 180 \times 5 = 900 = (2^2 \times 3^2 \times 5^2)$

Calculate the Quotient when M is divided by 25

Now divide the smallest perfect square $M = 900$ by $25$ to obtain the quotient $Q$.

$Q = \frac{900}{25} = 36$

Calculate the Sum of Digits of the Quotient

The quotient obtained is $36$. Summing its individual digits yields the final answer.

$\text{Sum of digits} = 3 + 6 = 9$