Step 1: Find prime factorization of each number

Express each number as a product of its prime factors: $24 = 2^3 \times 3$, $42 = 2 \times 3 \times 7$, and $56 = 2^3 \times 7$.

$24 = 2^3 \times 3,\quad 42 = 2 \times 3 \times 7,\quad 56 = 2^3 \times 7$

Step 2: Select the highest power of each prime factor

The prime factors involved are $2$, $3$, and $7$. Their highest powers appearing in the factorizations are $2^3$, $3^1$, and $7^1$.

$\text{Highest powers: } 2^3, 3, 7$

Step 3: Calculate the LCM

Multiply the highest powers of all prime factors together: $\text{LCM} = 2^3 \times 3 \times 7 = 8 \times 3 \times 7 = 168$.

$\text{LCM} = 8 \times 3 \times 7 = 168$