Determine the smallest 3 digit number which is exactly divisible by 6 8 12.

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**.

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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.

Learn how to find the smallest 3-digit number that is exactly divisible by 6, 8, and 12 with a step-by-step solution. This problem involves calculating the Least Common Multiple (LCM) and understanding divisibility rules. Suitable for students in Grades 5-7.

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