Finding the Least Common Multiple (LCM) of 48 and 54

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the numbers. To find the LCM of 48 and 54, we can use the prime factorization method.

Step-by-Step LCM Calculation

First, let's find the prime factorization of each number:

• For 48:
  • $48 = 2 \times 24$
  • $24 = 2 \times 12$
  • $12 = 2 \times 6$
  • $6 = 2 \times 3$
  • So, the prime factorization of 48 is $2 \times 2 \times 2 \times 2 \times 3$, which can be written as $2^4 \times 3^1$.
• For 54:
  • $54 = 2 \times 27$
  • $27 = 3 \times 9$
  • $9 = 3 \times 3$
  • So, the prime factorization of 54 is $2 \times 3 \times 3 \times 3$, which can be written as $2^1 \times 3^3$.

Now, we find the LCM by taking the highest power of each prime factor that appears in either factorization.

• The prime factors are 2 and 3.
• Highest power of 2: $2^4$ (from 48)
• Highest power of 3: $3^3$ (from 54)

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

Calculate the values:

• $2^4 = 2 \times 2 \times 2 \times 2 = 16$
• $3^3 = 3 \times 3 \times 3 = 27$

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

To calculate $16 \times 27$:

  27
x 16
----
 162 (27 x 6)
 270 (27 x 10)
----
 432

So, the LCM of 48 and 54 is 432.

Evaluating the Given Options

Let's calculate the value for each option provided:

• Option 1: $6 \times 8 \times 9$
  • $6 \times 8 = 48$
  • $48 \times 9 = 432$
• Option 2: $48 \times 54$
  • $48 \times 54 = 2592$
• Option 3: $6 \times 2 \times 9$
  • $6 \times 2 = 12$
  • $12 \times 9 = 108$
• Option 4: $6 \times 8 \times 3$
  • $6 \times 8 = 48$
  • $48 \times 3 = 144$

Conclusion

Comparing the calculated LCM (432) with the values of the options, we see that Option 1 ($6 \times 8 \times 9$) equals 432. Therefore, the expression $6 \times 8 \times 9$ represents the LCM of 48 and 54.

Revision Table: LCM of 48 and 54

LCM(48, 54) = $2^4 \times 3^3 = 16 \times 27 = 432$

Additional Information on LCM and Prime Factorization

The Least Common Multiple (LCM) is a fundamental concept in arithmetic and number theory. It is particularly useful when adding or subtracting fractions with different denominators, as the LCM of the denominators is the least common denominator (LCD).

The prime factorization method is a reliable way to find the LCM of any set of positive integers. By breaking down each number into its prime factors, we can systematically determine the smallest number that contains all the required prime factors with their necessary powers.

Remember that the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) is another related concept. While LCM uses the highest powers of all prime factors, GCD uses the lowest powers of only the common prime factors.