Problem 21 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 6 and 12

Step-by-Step Solution

Answer

The least common multiple of 6 and 12 is 12.

Step 1: List the Factors

For each number, we need to list their prime factors. For 6, the prime factors are 2 and 3, which can be written as \(6 = 2 \times 3\). For 12, the prime factors are 2, 2, and 3, which can be written as \(12 = 2^2 \times 3\).

Step 2: Identify the Highest Powers

Next, identify the highest power of each prime number present in the factorizations from Step 1. For the prime number 2, the highest power is \(2^2\) (from 12). For the prime number 3, the highest power is \(3^1\) (from both 6 and 12).

Step 3: Calculate the LCM

To find the least common multiple, we multiply the highest powers of all the prime numbers identified: \(2^2 \times 3^1 = 4 \times 3 = 12\).

Key Concepts

Prime Factors

Prime factors are the building blocks of numbers. They are the prime numbers that can be multiplied together to give the original number. A prime number is one that is only divisible by 1 and itself, such as 2, 3, 5, 7, and so on. To break a number down into its prime factors, you need to divide it by the smallest prime number possible until you cannot divide anymore.
For example, for the number 6, the prime factors are 2 and 3, because when multiplied together, they give the original number:

• 2 * 3 = 6

Similarly, for the number 12, the prime factors are 2, 2, and 3 which can be represented as:

• 2 * 2 * 3 = 12

This means 12 can be written as 2 squared times 3, which is expressed as \(12 = 2^2 \times 3\).
Calculating the prime factors is the first step in finding the Least Common Multiple (LCM) or even solving other number-based problems.

Highest Powers

When calculating the least common multiple, identifying the highest power of each prime factor from different numbers is crucial. We look for the largest exponent that appears in all the factorizations.
Taking our earlier example, consider the number 6, with prime factors \(6 = 2 \times 3\), and the number 12, with prime factors \(12 = 2^2 \times 3\).

• For the prime factor 2, we have different powers: 2 and \(2^2\).
• The highest power of 2 here is \(2^2\).
• For the prime factor 3, both numbers have \(3^1\).
• So the highest power of 3 is \(3^1\).

These 'highest powers' reflect the most significant role each prime factor plays in forming the smallest multiple shared by both numbers.
This means we will use these highest powers to find the LCM.

Multiplication of Prime Factors

Now that we have identified which prime factors have the highest powers, the next step is to multiply these together to determine the least common multiple.

• Take the highest power of the prime factor 2, which is \(2^2\).
• Then, take the highest power of the prime factor 3, which is \(3^1\).

Now, we multiply these highest powers together:

• \(2^2 \times 3^1\)

When calculated, this becomes 4 multiplied by 3, which equals 12. Thus, the least common multiple of the numbers 6 and 12 is 12.
This process ensures that the LCM is the smallest number that both original numbers can divide without leaving a remainder.

Factors of Numbers

Factors are numbers that you can multiply together to get another number. Understanding how to break down a number into its factors is key to solving problems involving the least common multiple or its counterpart, the greatest common divisor.
For instance, the factors of 6 are:

• 1, 2, 3, and 6.

Similarly, the factors of 12 are:

• 1, 2, 3, 4, 6, and 12.

Being able to list out the full set of factors for any number allows you to better understand its divisibility and relationship to other numbers.
Identifying these factors assists in finding not just the least common multiple but also in various math problems requiring divisibility considerations. Being familiar with factors can make complex calculations a breeze!