The smallest positive number divisible by all integers from 1 to 9 is 2520.
Understanding Divisibility
Divisibility refers to whether one number can be divided by another number with no remainder. When looking for a number divisible by a range of integers, we are essentially finding a common multiple of those integers.
The Least Common Multiple (LCM)
The smallest number divisible by all the integers from 1 to 9 is known as their Least Common Multiple (LCM). The LCM is calculated by finding the prime factorization of each number and then taking the highest power of each prime that appears in the factorizations.
Calculation Breakdown
Here's a breakdown of the prime factorization of numbers 1 through 9 to determine the LCM, which is 2520:
- 1: 1
- 2: 2
- 3: 3
- 4: 22
- 5: 5
- 6: 2 * 3
- 7: 7
- 8: 23
- 9: 32
To find the LCM, we use the highest power of each unique prime factor:
23 32 5 7 = 8 9 5 7 = 2520
Key Takeaway
- 2520 is the smallest positive number that is divisible by all integers from 1 to 9.
- The number is the Least Common Multiple (LCM) of 1 through 9
- There is no smaller number that satisfies this divisibility criteria.
