To find the smallest 3-digit number exactly divisible by 6, 8, and 12, we can break the problem down into the following steps:

Step 1: Determine the Least Common Multiple (LCM) of 6, 8, and 12.

1. Prime factorization:

   • $6 = 2 \times 3$
   • $8 = 2^3$
   • $12 = 2^2 \times 3$

2. LCM Calculation: The LCM is found by taking the highest power of each prime factor that appears in any of the factorizations:

   • The highest power of 2 is $2^3$.
   • The highest power of 3 is $3^1$.

   Therefore, the LCM is: $\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24$

Step 2: Find the Smallest 3-Digit Number Divisible by 24.

The smallest 3-digit number is 100. To determine the smallest number divisible by 24, we divide 100 by 24:

$\frac{100}{24} \approx 4.17$

This result tells us that the smallest multiple of 24 greater than 100 is $5 \times 24$.

$5 \times 24 = 120$

Conclusion:

The smallest 3-digit number that is exactly divisible by 6, 8, and 12 is 120.

---

Would you like more details or have any other questions?

Here are 5 related questions to explore:

1. What is the LCM of other common sets of numbers?
2. How is the greatest common divisor (GCD) calculated?
3. How do you determine if a number is divisible by another without a calculator?
4. Can you find the smallest 4-digit number divisible by 6, 8, and 12?
5. What is the relationship between LCM and GCD?

Tip: Remember that finding the LCM involves taking the highest powers of all primes involved in the factorization.