What is the least common multiple of 84 and 56?

Steps to Solve

1. Find the Prime Factorization of Each Number

Start by determining the prime factors of 84 and 56.

For 84:

• Divide by 2: $84 \div 2 = 42$
• Divide by 2 again: $42 \div 2 = 21$
• Divide by 3: $21 \div 3 = 7$
• 7 is a prime number and cannot be divided further.

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

For 56:

• Divide by 2: $56 \div 2 = 28$
• Divide by 2 again: $28 \div 2 = 14$
• Divide by 2 again: $14 \div 2 = 7$
• 7 is a prime number and cannot be divided further.

So, the prime factorization of 56 is: $$ 56 = 2^3 \times 7^1 $$

2. Identify the Highest Powers of Each Prime Factor

Next, identify the highest powers of all prime factors that appear in the factorizations of both numbers.

• The prime factor 2: Highest power is $2^3$ (from 56)
• The prime factor 3: Highest power is $3^1$ (from 84)
• The prime factor 7: Highest power is $7^1$ (from both)

3. Calculate the Least Common Multiple (LCM)

Combine the highest powers of each prime factor to find the LCM.

Therefore, $$ LCM(84, 56) = 2^3 \times 3^1 \times 7^1 $$

Now, let's compute that:

$$ LCM(84, 56) = 8 \times 3 \times 7 $$

Calculating step-by-step:

First compute $8 \times 3 = 24$.

Then compute $24 \times 7 = 168$.

4. Final Result

The least common multiple of 84 and 56 is: $$ LCM(84, 56) = 168 $$