The smallest number divisible by all numbers between 5 and 15 is 360360.
Understanding the Problem
We need to find the Least Common Multiple (LCM) of the numbers 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The LCM is the smallest number that all of these given numbers divide into evenly.
Calculation
According to the provided reference, the least number divisible by all the natural numbers from 5 to 15 is 360360. This was determined by calculating the LCM of these numbers. Here's a breakdown of why that number is the correct answer:
- Prime Factorization: To find the LCM, we need the prime factorization of each number.
- Highest Powers: We then take the highest power of each prime factor that appears in any of the factorizations.
- Multiplication: Multiply these highest powers together to get the LCM.
Let's look at some examples of the numbers in between:
| Number | Prime Factorization |
|---|---|
| 5 | 5 |
| 6 | 2 x 3 |
| 7 | 7 |
| 8 | 23 |
| 9 | 32 |
| 10 | 2 x 5 |
| 11 | 11 |
| 12 | 22 x 3 |
| 13 | 13 |
| 14 | 2 x 7 |
| 15 | 3 x 5 |
To find the LCM, we will use the highest power of each prime factor. Those factors are: 23, 32, 5, 7, 11 and 13. Therefore, our calculation is 23 x 32 x 5 x 7 x 11 x 13 = 8 x 9 x 5 x 7 x 11 x 13 = 360360.
- Verification: If you divide 360360 by any number between 5 to 15, there will be no remainder. This verifies that it is the smallest number divisible by the requested set.
Conclusion
Based on the reference provided, the smallest number divisible by all numbers between 5 and 15 is indeed 360360.
