What is the lowest common multiple of 28 and 24?

Steps to Solve

1. Prime Factorization of 28

First, we find the prime factorization of 28.

The factors of 28 are: $$ 28 = 2 \times 14 $$ $$ 14 = 2 \times 7 $$

Thus, the prime factorization of 28 is: $$ 28 = 2^2 \times 7^1 $$

2. Prime Factorization of 24

Next, we find the prime factorization of 24.

The factors of 24 are: $$ 24 = 2 \times 12 $$ $$ 12 = 2 \times 6 $$ $$ 6 = 2 \times 3 $$

Thus, the prime factorization of 24 is: $$ 24 = 2^3 \times 3^1 $$

3. Identifying Maximum Powers of Prime Factors

Now we identify the maximum powers of all prime factors from both factorizations.

For the primes in both factorizations:

• The maximum power of 2 is $2^3$ from 24.
• The maximum power of 3 is $3^1$ from 24.
• The maximum power of 7 is $7^1$ from 28.

4. Calculating the LCM

Using the maximum powers, we can calculate the LCM:

$$ \text{LCM} = 2^3 \times 3^1 \times 7^1 $$

Now calculate the LCM: $$ 2^3 = 8 $$ $$ 3^1 = 3 $$ $$ 7^1 = 7 $$

Now, multiply these results together:

$$ 8 \times 3 = 24 $$ $$ 24 \times 7 = 168 $$

Thus, the LCM is: $$ \text{LCM}(28, 24) = 168 $$