Question

Find the least common multiple of each pair of numbers. 38 and 190

Short answer

The LCM of 38 and 190 is 190.

Step-by-step solution

- Find the Prime Factorization

First, determine the prime factors of each number. For 38: 38 = 2 × 19. For 190: 190 = 2 × 5 × 19.

- Identify All Prime Factors

List all the prime factors found: 2, 5, 19.

- Determine the Highest Power of Each Prime Factor

Look at each prime factor and choose the highest power that appears in each number. Here, all prime factors already appear at their highest power: 2^1, 5^1, 19^1.

- Calculate the LCM

Multiply these highest powers together to find the least common multiple: LCM = 2^1 × 5^1 × 19^1 = 2 × 5 × 19 = 190.

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its basic building blocks: prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, let's look at the given exercise. The number 38 breaks down into the primes 2 and 19, because 38 = 2 × 19. Similarly, 190 can be broken down into 2, 5, and 19, because 190 = 2 × 5 × 19.
By understanding the prime factors of a number, you simplify it into its core pieces, which helps in various calculations including finding the Least Common Multiple (LCM).

Prime Factors

Prime factors are the prime numbers that multiply together to give a composite number. Identifying them is a key step in solving various mathematical problems. For our exercise, we identified the prime factors of 38 and 190:

• 38: 2, 19
• 190: 2, 5, 19

Notice that both numbers share the prime factors 2 and 19. Prime factors help you understand the common building blocks of numbers, which is crucial for calculating the LCM.

LCM Calculation

To calculate the Least Common Multiple (LCM) of two numbers, you need to multiply the highest power of each prime factor found in the numbers. In our exercise, we list all prime factors: 2, 5, and 19. Each prime factor appears at its highest power of 1. So, the LCM calculation looks like this: \[ LCM = 2^1 \times 5^1 \times 19^1 = 2 \times 5 \times 19 = 190 \] By multiplying these highest powers, we obtain the LCM, which is 190. This method ensures that the LCM is the smallest number that both 38 and 190 can divide into without leaving a remainder.

Mathematics

Mathematics involves various techniques and processes for solving problems effectively. Learning concepts like prime factorization and calculating LCMs helps in understanding how numbers work at a deeper level. These skills are not only useful in academic settings but also in everyday problem-solving.

• Prime factorization teaches you to break down complex problems into simpler parts.
• Knowing prime factors gives insights into the composition of numbers.
• Understanding LCM is crucial for tasks involving synchronization of events, comparison of different cycles, etc.

By mastering these foundational concepts, you can tackle more complex mathematical problems with confidence.