lcm of 56 and 84

Steps to Solve

1. Prime Factorization of 56

To find the prime factorization of 56, we can divide it by the smallest prime number, which is 2.

$$ 56 \div 2 = 28 \ 28 \div 2 = 14 \ 14 \div 2 = 7 \ 7 \div 7 = 1 $$

Thus, the prime factorization of 56 is $2^3 \times 7^1$.

2. Prime Factorization of 84

Next, we find the prime factorization of 84, starting again with the smallest prime number.

$$ 84 \div 2 = 42 \ 42 \div 2 = 21 \ 21 \div 3 = 7 \ 7 \div 7 = 1 $$

So, the prime factorization of 84 is $2^2 \times 3^1 \times 7^1$.

3. Identify the Highest Powers of Each Prime

Now, we take the highest power of each prime factor from both factorizations.

For 2, the highest power is $2^3$.
For 3, the highest power is $3^1$.
For 7, the highest power is $7^1$.

4. Calculate the LCM

The LCM is found by multiplying these highest powers together:

$$ LCM = 2^3 \times 3^1 \times 7^1 $$

Compute this product:

$$ LCM = 8 \times 3 \times 7 $$

5. Final Calculation

Now we calculate the result step by step:

First calculate $8 \times 3$:

$$ 8 \times 3 = 24 $$

Then multiply by 7:

$$ 24 \times 7 = 168 $$

Thus, the LCM of 56 and 84 is 168.