The smallest number that is divisible by all integers from 1 to 9 is known as the least common multiple (LCM) of these numbers. To find this, we need to determine the highest powers of the prime factors for each number from 1 to 9.
The prime factorization of each number is as follows:
- 1 = 1
- 2 = 2
- 3 = 3
- 4 = 22
- 5 = 5
- 6 = 2 × 3
- 7 = 7
- 8 = 23
- 9 = 32
The LCM will include each prime number raised to the highest power that appears in these factorizations:
- For the prime number 2, the highest power is 23 (from 8).
- For the prime number 3, the highest power is 32 (from 9).
- For the prime number 5, the highest power is 51 (from 5).
- For the prime number 7, the highest power is 71 (from 7).
Now we can calculate the LCM by multiplying these together:
LCM = 23 × 32 × 51 × 71 = 8 × 9 × 5 × 7
Calculating this step-by-step:
- 8 × 9 = 72
- 72 × 5 = 360
- 360 × 7 = 2520
Therefore, the smallest number that is divisible by all integers from 1 through 9 is 2520.