Question

Find the least common multiple of the given numbers. \(11,13,\) and 17

Short answer

The LCM of 11, 13, and 17 is 2431.

Step-by-step solution

Identify the numbers

The given numbers are 11, 13, and 17.

Understand the concept of LCM

The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers.

Check if the numbers are prime

Check whether 11, 13, and 17 are prime numbers. Since each of these numbers is only divisible by 1 and itself, they are all prime numbers.

Calculate the LCM of prime numbers

If the numbers are all prime, the LCM is simply the product of the numbers. Hence, calculate the product: \[ 11 \times 13 \times 17 \]

Perform the multiplication

Multiply the numbers together: \[ 11 \times 13 = 143 \]\[ 143 \times 17 = 2431 \]

Write the final answer

The least common multiple of 11, 13, and 17 is 2431.

Key concepts

Prime Numbers

Prime numbers are the foundation of many mathematical concepts, including LCM calculation. A prime number is defined as a number greater than 1 that has no divisors other than 1 and itself. For instance:

• 11 is only divisible by 1 and 11
• 13 is only divisible by 1 and 13
• 17 is only divisible by 1 and 17

These characteristics make prime numbers significant in finding the least common multiple (LCM) because they simplify the process. Remember, identifying whether numbers are prime is crucial before performing LCM calculations, especially when dealing with smaller sets of prime numbers.

LCM Calculation

The least common multiple (LCM) is the smallest positive number that all given numbers can divide without leaving a remainder. Here's a step-by-step breakdown:

• Identify the numbers you are working with. In this case: 11, 13, and 17.
• Determine if the numbers are prime (as explained earlier).
• For prime numbers, the LCM is simply their product because they have no other common factors.

For our numbers, calculate:\[ LCM = 11 \times 13 \times 17 \text{.} \]The product of prime numbers is simplified because you're dealing with distinct factors that cannot overlap. This fundamental principle makes LCM calculation straightforward when all the numbers involved are prime.

Multiplication

Multiplication is a key operation in mathematics, often used in calculating the LCM. To find the LCM of given prime numbers, you multiply them together. Here's how:

• First, multiply 11 by 13, yielding: \[ 11 \times 13 = 143 \]
• Next, take the result (143) and multiply it by 17: \[ 143 \times 17 = 2431 \]
• Therefore, the LCM of 11, 13, and 17 is 2431.

This process highlights how understanding basic multiplication can assist you in tackling more complex problems, like finding the least common multiple. Breaking multiplication into smaller, manageable steps makes the problem easier to handle and verify.