what is lcm
What is LCM?
Answer:
The Least Common Multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers. Understanding the concept of LCM is essential in various fields of mathematics and its applications. Here’s a detailed explanation:
**1. Definition and Importance:
- The LCM of two numbers (a) and (b) is the smallest positive integer (m) such that (m) is divisible by both (a) and (b). This means:
\text{LCM}(a, b) = m \quad \text{where} \quad m \geq \max(a, b) \ \text{and} \ m \mod a = 0 \ \text{and} \ m \mod b = 0. - The concept is important in solving problems involving addition or subtraction of fractions with different denominators, scheduling problems, and cryptographic algorithms among others.
**2. Methods to Find LCM:
- Prime Factorization Method:
1. Find the prime factors of each number.
- Example: Find the LCM of 12 and 18.
- Prime factorization of (12) is 2^2 \times 3^1
- Prime factorization of (18) is 2^1 \times 3^2.
2. Take the highest power of each prime factor.
- For (2), the highest power is (2^2).
- For (3), the highest power is (3^2).
3. Multiply these highest powers together:
- 2^2 \times 3^2 = 4 \times 9 = 36.
-
So, \text{LCM}(12, 18) = 36.
- Division Method:
1. Write the numbers in a row and divide by the smallest prime number that divides at least one of the numbers.
- Continue the process until only 1s remain.
2. Example: Find the LCM of 8 and 14:
\begin{array}{cccc}
2 & | & 8 & 14 \\
2 & | & 4 & 7 \\
2 & | & 2 & 7 \\
7 & | & 1 & 7 \\
1 & & 1 & 1 \\
\end{array}
3. Multiply all the divisors:
2 \times 2 \times 2 \times 7 = 56
So, $\text{LCM}(8, 14) = 56$.
- Using GCD (Greatest Common Divisor) Method:
1. Use the relationship between LCM and GCD:
\text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)}
2. **Example: Find the LCM of 8 and 12, given $\text{GCD}(8, 12) $=
\text{LCM}(8, 12) = \frac{|8 \times 12|}{4} = 24
Final Answer:
The Least Common Multiple (LCM) is the smallest positive integer that is exactly divisible by each of the given numbers. The methods to find it include prime factorization, division method, and using the GCD. Understanding and calculating LCM is fundamental in various mathematical operations and applications.