The least number which when divided by 12, 18, and 27 leaves a remainder of 9 is:

To find this number, we must first find the Least Common Multiple (LCM) of the divisors (12, 18, 27) and then add the required remainder 9.

Step 1: To find the LCM of 12, 18, and 27

We use prime factorisation to find the LCM:

• $$12=2^2\times3^1$$
• $$18 = 2^1 \times 3^2$$
• $$27 = 3^3$$

The LCM is the product of the highest power of all prime factors:

$$\text{LCM}(12, 18, 27) = 2^2 \times 3^3$$

$$\text{LCM} = 4 \times 27$$

$$\text{LCM} = 108$$

Step 2: Add the Remainder

The required number is (LCM) + (Remainder).

$$\text{Required Number} = 108 + 9$$

$$\text{Required Number} = 117$$

Hence, the correct answer is option A.