The smallest number divisible by both 9 and 12 is 36.
Understanding the Least Common Multiple (LCM)
The question asks for the smallest number that both 9 and 12 can divide into without leaving a remainder. This is known as the Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of two or more given integers.
Here's a brief breakdown using the given reference:
According to the provided information, the LCM of 9 and 12 is 36. This is the smallest number that is evenly divisible by both 9 and 12.
How to Find the LCM:
Several methods can be used to find the LCM. Here are a few examples:
- Listing Multiples: You could list out multiples of each number until you find a common one.
- Multiples of 9: 9, 18, 27, 36, 45...
- Multiples of 12: 12, 24, 36, 48...
- The first common multiple in both lists is 36.
- Prime Factorization: Another method involves breaking down the numbers into prime factors.
- Prime factors of 9: 3 x 3 or 3²
- Prime factors of 12: 2 x 2 x 3 or 2² x 3
- To find the LCM, take the highest power of each prime factor that appears: 2² x 3² = 4 x 9 = 36.
- Using a Formula: For two numbers, you can use the formula: LCM (a, b) = (|a b|) / GCD(a, b), where GCD is the Greatest Common Divisor. The GCD of 9 and 12 is 3. So, LCM(9,12) = (9 12)/3 = 108 / 3 = 36.
Example:
Let's check that 36 is divisible by both 9 and 12:
- 36 ÷ 9 = 4
- 36 ÷ 12 = 3
As you can see, 36 is perfectly divisible by both numbers without a remainder, and it’s the smallest number with this property.
Conclusion
Therefore, based on the information provided and established mathematical principles, the smallest number divisible by both 9 and 12 is 36.
