Understanding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the numbers. Finding the LCM is useful in various mathematical problems, such as adding or subtracting fractions with different denominators. We need to find the Least Common Multiple of the numbers 48 and 54.

Methods to Find the LCM

There are several ways to find the Least Common Multiple. Two common methods are:

• Listing Multiples: Write down the multiples of each number until a common multiple is found.
• Prime Factorization Method: Find the prime factors of each number and use them to calculate the LCM.

We will use the Prime Factorization Method as it is generally more efficient for larger numbers.

Finding the Prime Factors of 48 and 54

First, let's break down each number into its prime factors.

Prime Factorization of 48

We divide 48 by the smallest prime numbers repeatedly:

• \(48 \div 2 = 24\)
• \(24 \div 2 = 12\)
• \(12 \div 2 = 6\)
• \(6 \div 2 = 3\)
• \(3 \div 3 = 1\)

So, the prime factorization of 48 is \(2 \times 2 \times 2 \times 2 \times 3\), which can be written in exponential form as \(2^4 \times 3^1\).

Prime Factorization of 54

Now, let's find the prime factors of 54:

• \(54 \div 2 = 27\)
• \(27 \div 3 = 9\)
• \(9 \div 3 = 3\)
• \(3 \div 3 = 1\)

So, the prime factorization of 54 is \(2 \times 3 \times 3 \times 3\), which can be written in exponential form as \(2^1 \times 3^3\).

Calculating the LCM using Prime Factors

To find the Least Common Multiple (LCM) using the prime factorization method, we follow these steps:

1. List all the prime factors that appear in the factorization of any of the numbers. In this case, the prime factors are 2 and 3.
2. For each prime factor, find the highest power that appears in the factorization of either number.
   • For the prime factor 2: The powers are \(2^4\) (from 48) and \(2^1\) (from 54). The highest power is \(2^4\).
   • For the prime factor 3: The powers are \(3^1\) (from 48) and \(3^3\) (from 54). The highest power is \(3^3\).
3. Multiply these highest powers together to get the LCM.

LCM \((48, 54) = (\text{Highest power of 2}) \times (\text{Highest power of 3})\)

LCM \((48, 54) = 2^4 \times 3^3\)

Calculate the values of these powers:

• \(2^4 = 2 \times 2 \times 2 \times 2 = 16\)
• \(3^3 = 3 \times 3 \times 3 = 27\)

Now, multiply these results:

LCM \((48, 54) = 16 \times 27\)

Let's perform the multiplication:

\(16 \times 27 = 16 \times (20 + 7) = (16 \times 20) + (16 \times 7) = 320 + 112 = 432\)

So, the Least Common Multiple of 48 and 54 is 432.

To find the LCM, take the highest power of each prime factor present:

• Highest power of 2: \(2^4\)
• Highest power of 3: \(3^3\)

LCM = \(2^4 \times 3^3 = 16 \times 27 = 432\).

Verification (Optional)

We can verify that 432 is indeed a multiple of both 48 and 54:

• \(432 \div 48 = 9\) (\(48 \times 9 = (50-2) \times 9 = 450 - 18 = 432\))
• \(432 \div 54 = 8\) (\(54 \times 8 = (50+4) \times 8 = 400 + 32 = 432\))

Since 432 is divisible by both 48 and 54, it is a common multiple. To be the Least Common Multiple, it must be the smallest such positive number, which the prime factorization method guarantees by using the minimum set of prime factors raised to their necessary powers.

Conclusion

The Least Common Multiple of 48 and 54 is 432.

Revision Table: Key Concepts

Additional Information: Relation between LCM and GCF

For any two positive integers, say 'a' and 'b', there is a relationship between their LCM and GCF:

LCM\((a, b) \times\) GCF\((a, b) = a \times b\)

Let's find the GCF of 48 and 54 using prime factorization:

• \(48 = 2^4 \times 3^1\)
• \(54 = 2^1 \times 3^3\)

GCF takes the lowest power of common prime factors. Common prime factors are 2 and 3.

• Lowest power of 2: \(2^1\)
• Lowest power of 3: \(3^1\)

GCF \((48, 54) = 2^1 \times 3^1 = 2 \times 3 = 6\)

Now, let's check the relationship:

LCM \((48, 54) \times\) GCF \((48, 54) = 432 \times 6\)

\(432 \times 6 = 2592\)

And \(a \times b = 48 \times 54\)

\(48 \times 54 = 2592\)

The relationship holds true: \(432 \times 6 = 48 \times 54 = 2592\). This confirms our calculated LCM and GCF are correct.