LCM of 48 and 64

Find the Least Common Multiple (LCM) of 48 and 64. The LCM is the smallest number that both 48 and 64 divide evenly.

Answer: LCM(48, 64) = 192

Step-by-Step Solution

Step 1 List the Multiples of 48

To find the LCM of 48 and 64, we first list the multiples of each number.

Multiples of 48 are: 48 × 1, 48 × 2, 48 × 3, ...

Multiples of 48: 48, 96, 144, 192, 240, 288, ...

Step 2 List the Multiples of 64

Now we list the multiples of 64:

Multiples of 64: 64, 128, 192, 256, 320, ...

Step 3 Find the Least Common Multiple

The common multiples are numbers that appear in both lists.

The smallest common multiple is the LCM.

Looking at both lists, the first number that appears in both is 192.

LCM(48, 64) = 192

Step 4 Prime Factorization Method

We can also find the LCM using prime factorization:

48 = 2^4 × 3

64 = 2^6

Take each prime factor with the highest power:

LCM = 2^6 × 3 = 192

Step 5 Using the GCF Formula

The LCM can be calculated using the GCF with this formula:

LCM(a, b) = (a × b) ÷ GCF(a, b)

GCF(48, 64) = 16

LCM = (48 × 64) ÷ 16 = 3072 ÷ 16 = 192

Step 6 Final Answer

LCM(48, 64) = 192