Calculate the LCM

To find the numbers divisible by both 8 and 12, we must first find their Least Common Multiple (LCM). The prime factorization is $8 = 2^3$ and $12 = 2^2 \times 3$. The LCM is $2^3 \times 3 = 24$.

$LCM(8, 12) = 24$

Identify multiples of 24

The numbers divisible by both 8 and 12 are multiples of 24: $24, 48, 72, 96, 120, 144, \dots$

$24 \times 1 = 24, 24 \times 2 = 48, 24 \times 3 = 72, 24 \times 4 = 96, 24 \times 5 = 120$

Select the smallest 3-digit number

Looking at the multiples, 96 is the largest 2-digit multiple, and 120 is the smallest 3-digit multiple of 24.

$120$