Step 1: Find Prime Factorization

Find the prime factors of each given number: $24 = 2^3 \times 3$, $40 = 2^3 \times 5$, and $56 = 2^3 \times 7$.

$24 = 2^3 \times 3, \quad 40 = 2^3 \times 5, \quad 56 = 2^3 \times 7$

Step 2: Calculate the LCM

The LCM is found by taking the highest power of each prime factor involved: $2^3 \times 3 \times 5 \times 7 = 840$.

$\text{LCM} = 2^3 \times 3 \times 5 \times 7 = 840$

Step 3: Find Smallest 4-Digit Multiple

Divide the smallest 4-digit number (1000) by the LCM (840) to find the multiplier: $1000 / 840 \approx 1.19$. Round up to the next whole number, which is 2.

$k = \lceil 1000 / 840 \rceil = 2$

Step 4: Compute the Final Number

Multiply the LCM by 2 to get the smallest 4-digit number divisible by 24, 40, and 56: $840 \times 2 = 1680$.

$840 \times 2 = 1680$