what is the lcm of 55 and 137

Natashya

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23-12-2024

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The least common multiple (LCM) is the smallest positive integer that is divisible by both numbers without leaving a remainder. Finding the LCM of 55 and 137 is a straightforward process, but understanding the method is key. Let's explore how to calculate it.

Understanding Least Common Multiple (LCM)

Before we dive into calculating the LCM of 55 and 137, let's briefly review what an LCM is. The LCM is crucial in various mathematical applications, including simplifying fractions and solving problems related to cycles and periodic events.

• Divisibility: A number is divisible by another if it can be divided evenly, with no remainder.
• Common Multiple: A multiple is a number obtained by multiplying a given number by an integer. A common multiple is a number divisible by both of your given numbers.
• Least Common Multiple (LCM): The smallest of all the common multiples is the LCM.

Methods for Finding the LCM of 55 and 137

There are several ways to determine the least common multiple of two numbers. We'll focus on two common methods:

Method 1: Prime Factorization

This method is particularly useful for larger numbers. It involves breaking down each number into its prime factors (prime numbers that multiply to give the original number).

1. Prime Factorization of 55: 55 = 5 x 11

2. Prime Factorization of 137: 137 is a prime number, meaning its only factors are 1 and itself.

3. Finding the LCM: To find the LCM using prime factorization, take the highest power of each prime factor present in the factorizations of both numbers and multiply them together. In this case: LCM(55, 137) = 5 x 11 x 137 = 7535

Therefore, the LCM of 55 and 137 is 7535.

Method 2: Using the Formula (For Smaller Numbers)

A simpler method exists for smaller numbers. It utilizes the relationship between the LCM and the greatest common divisor (GCD) of two numbers:

• Formula: LCM(a, b) = (|a x b|) / GCD(a, b)

Where:

• a and b are the two numbers.
• GCD(a, b) is the greatest common divisor of a and b (the largest number that divides both a and b without a remainder).

1. GCD of 55 and 137: Since 137 is a prime number and doesn't divide 55, the GCD(55, 137) = 1.

2. Applying the Formula: LCM(55, 137) = (55 x 137) / 1 = 7535

Again, the LCM of 55 and 137 is 7535.

Conclusion

The least common multiple of 55 and 137 is 7535. Both the prime factorization method and the formula method confirm this result. Choosing the best method depends on the numbers involved; for larger numbers, prime factorization might be more efficient. Understanding these methods helps you tackle various LCM problems efficiently.