Final answer:
The smallest number that has 10, 15, and 6 as factors is calculated by finding the least common multiple (LCM) of these numbers, which is 30.
Step-by-step explanation:
To find the smallest number that has the factors 10, 15, and 6, you need to calculate the least common multiple (LCM) of these numbers. This is because the LCM is the smallest number that all of the factors can divide without leaving a remainder.
First, break down each number into its prime factors:
- 10 is 2 × 5
- 15 is 3 × 5
- 6 is 2 × 3
Then, identify the highest powers of all prime factors present:
- For 2: The highest power is 21 (present in 10 and 6)
- For 3: The highest power is 31 (present in 15 and 6)
- For 5: The highest power is 51 (present in 10 and 15)
The LCM is the product of these highest powers: LCM = 21 × 31 × 51 = 2 × 3 × 5 = 30.
Therefore, the smallest number that has 10, 15, and 6 as factors is 30. This number can be evenly divided by all three given factors.