The smallest number divisible by both 11 and 12 is 132.
Understanding the Least Common Multiple (LCM)
The question asks for the smallest number that can be divided by both 11 and 12 without leaving a remainder. This value is mathematically known as the Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of two or more given integers. It is a fundamental concept in arithmetic, especially useful when working with fractions to find a common denominator or when dealing with problems involving cycles.
To learn more about the Least Common Multiple, you can refer to educational resources like Khan Academy's introduction to LCM.
Methods to Determine the LCM
There are several straightforward methods to find the LCM of two numbers like 11 and 12.
Method 1: Listing Multiples
This method involves listing the multiples of each number until a common multiple is found. The first common multiple you encounter will be the LCM.
Let's list the first few multiples of 11 and 12:
| Multiples of 11 | Multiples of 12 |
|---|---|
| 11 | 12 |
| 22 | 24 |
| 33 | 36 |
| 44 | 48 |
| 55 | 60 |
| 66 | 72 |
| 77 | 84 |
| 88 | 96 |
| 99 | 108 |
| 110 | 120 |
| 121 | 132 |
| 132 | 144 |
By listing the multiples, we can see that 132 is the first number that appears in both lists. This means 132 is the smallest number exactly divisible by both 11 and 12.
Method 2: Prime Factorization
The prime factorization method involves breaking down each number into its prime factors. The LCM is then found by multiplying the highest power of all prime factors present in either number.
Here's how it works for 11 and 12:
-
Prime factorization of 11:
- 11 is a prime number, so its prime factorization is simply 11.
-
Prime factorization of 12:
- 12 = 2 × 6
- 12 = 2 × 2 × 3
- In exponential form, this is 2² × 3¹
-
Calculate the LCM:
- Take the highest power of each prime factor that appears in either factorization.
- For prime factor 2, the highest power is 2² (from 12).
- For prime factor 3, the highest power is 3¹ (from 12).
- For prime factor 11, the highest power is 11¹ (from 11).
- LCM (11, 12) = 2² × 3¹ × 11¹ = 4 × 3 × 11 = 12 × 11 = 132.
Why 132 is the Answer
The numbers 11 and 12 are relatively prime, meaning they do not share any common factors other than 1. When two numbers are relatively prime, their Least Common Multiple (LCM) is simply their product.
- 11 (prime number)
- 12 (factors: 1, 2, 3, 4, 6, 12)
Since there are no common prime factors between 11 and 12, their LCM is found by multiplying them together:
11 × 12 = 132
This confirms that 132 is indeed the smallest number divisible by both 11 and 12.
