The least common multiple (LCM) of the first ten natural numbers (1 through 10) is a key concept in number theory and has many practical applications in mathematics. To find the LCM, we need to determine the smallest number that is a multiple of all these integers.
First, let’s list the natural numbers we need to consider:
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9
- 10
Next, we can find the prime factorization of each of these numbers:
- 1 = 1
- 2 = 2
- 3 = 3
- 4 = 22
- 5 = 5
- 6 = 2 * 3
- 7 = 7
- 8 = 23
- 9 = 32
- 10 = 2 * 5
To compute the LCM, we will take the highest power of each prime number that appears in these factorizations:
- For prime 2: highest power is 23 (from 8)
- For prime 3: highest power is 32 (from 9)
- For prime 5: highest power is 51 (from 5 or 10)
- For prime 7: highest power is 71 (from 7)
Now, multiply these highest powers together:
- LCM = 23 * 32 * 51 * 71
- = 8 * 9 * 5 * 7
Now, calculate step by step:
- 8 * 9 = 72
- 72 * 5 = 360
- 360 * 7 = 2520
So, the least common multiple of the first 10 natural numbers is 2520.
This means that 2520 is the smallest number that can be divided evenly by each of the first ten natural numbers, making it a significant value in various mathematical applications.