Problem 85 - Algebra 1: Concepts and Skills

Question

Find the least common multiple of the numbers. 4 and 5

Step-by-Step Solution

Answer

The least common multiple of 4 and 5 is 20.

Step 1: Identifying the Numbers

The numbers given in this problem are 4 and 5

Step 2: Determining the LCM

Given that 4 and 5 are prime to each other (meaning, they have no common prime factors), their least common multiple can be found by simply multiplying them together. So, \`LCM(4, 5) = 4*5 = 20\`

Key Concepts

LCM calculation

When it comes to working with whole numbers, the term 'least common multiple' (LCM) is central for various mathematical concepts and operations, specifically when dealing with fractions or finding common denominators. The LCM is the smallest positive number that is a multiple of two or more numbers. To calculate the LCM of two numbers, follow these general steps:

• First, list the prime factors of each number.
• Next, identify the highest power of each prime number found in the factorization of each number.
• Finally, multiply these prime numbers together to get the LCM.

However, when two numbers are prime to each other—meaning they share no common prime factors—the LCM calculation simplifies to just multiplying the two numbers together, because each number is essentially a 'unique' prime factor in the equation. This is what happened with the numbers 4 and 5 in our exercise; since these numbers are coprime, their LCM is simply 4 multiplied by 5, which equals 20.

Prime factors

Prime factors are the building blocks of numbers, representing the most basic level of number division. A prime factor is a factor that is a prime number, which means it can only be divided by 1 and itself without leaving a remainder. To find the prime factors of a number, you'll need to perform prime factorization, which involves breaking down a number into a set of prime numbers that, when multiplied together, give the original number.
For instance, the number 4 can be broken down into prime factors of 2 and 2 (since 2 is a prime number and 2 x 2 = 4). This process requires a good understanding of prime numbers and comprehension of multiplication. Prime factorization is a critical step in various mathematical applications, including calculating the LCM, greatest common divisor (GCD), and simplifying fractions.

Multiplying prime numbers

When it comes to multiplying prime numbers, it's a smooth operation due to the nature of primes: since primes are numbers that are not divisible by any other number except 1 and themselves, multiplying them together will not create a number with complex factors. This makes the multiplication of prime numbers a straightforward arithmetical action with a clear outcome.

In our example involving the numbers 4 and 5, while 5 is a prime number, 4 is not. However, because 4 is comprised of a prime number (2) repeated, it behaves in a similar way when calculating the LCM. To fulfill the multiplication, simply take the prime number 5 and multiply it by each prime factor of 4, which yields (2 x 2 x 5). This gives us the product of 20, which is, as established, the LCM. Remember that the unique nature of prime numbers ensures that their products introduce no additional common factors besides 1 and the numbers themselves.