Is the LCM of two numbers always greater than the two numbers ?

Answer

To determine whether the LCM (Least Common Multiple) of two numbers is always greater than the two numbers, we can analyze this through examples and calculations.

### Step-by-Step Solution:

1. **Choose Two Numbers**: Let's take the numbers 15 and 30 as our example.

2. **Find the LCM Using Factorization Method**:
- Factor 15:
- 15 = 3 × 5
- Factor 30:
- 30 = 2 × 3 × 5
- Now, to find the LCM, we take the highest power of each prime factor:
- From 15: 3 (1 time), 5 (1 time)
- From 30: 2 (1 time), 3 (1 time), 5 (1 time)
- Therefore, the LCM is:
- LCM = 2^1 × 3^1 × 5^1 = 2 × 3 × 5 = 30

3. **Compare the LCM with the Original Numbers**:
- The LCM we calculated is 30.
- The two numbers we started with are 15 and 30.
- Here, we see that the LCM (30) is equal to one of the numbers (30).

4. **Conclusion**:
- Since the LCM (30) is not greater than both numbers (15 and 30), we conclude that the statement "the LCM of two numbers is always greater than the two numbers" is **not true**.