The smallest number divisible by 6, 8, and 9 is 72.
Let's break down how to find this:
Finding the Least Common Multiple (LCM)
To find the smallest number divisible by several numbers, we need to determine their Least Common Multiple (LCM). Here's how:
Prime Factorization
-
Find the prime factorization of each number:
- 6 = 2 x 3
- 8 = 2 x 2 x 2 = 2³
- 9 = 3 x 3 = 3²
-
Identify the highest power of each prime factor present:
- The highest power of 2 is 2³ (from 8).
- The highest power of 3 is 3² (from 9).
-
Multiply these highest powers together:
- LCM = 2³ x 3² = 8 x 9 = 72
Therefore, 72 is the smallest number divisible by 6, 8, and 9.
Reference Information
The reference provided states: "So 1008 is the smallest 4-digit divisible by 6, 8 and 9.08-Jul-2018". This informs us about the smallest four-digit number divisible by 6, 8, and 9, but doesn't directly address our initial question about the absolute smallest number divisible by these numbers.
Examples
- 72 / 6 = 12
- 72 / 8 = 9
- 72 / 9 = 8
Importance of LCM
The LCM is a fundamental concept in mathematics that helps in solving various problems:
- Fractions: Finding the lowest common denominator to add or subtract fractions.
- Scheduling: Determining when events will coincide.
- Number Theory: Solving divisibility problems.
