What is the least common multiple of 84 and 56?
Understand the Problem
The question is asking for the least common multiple (LCM) of the two numbers 84 and 56. To find the LCM, we can use the prime factorization method or the relationship between LCM and greatest common divisor (GCD).
Answer
The least common multiple (LCM) of 84 and 56 is $168$.
Answer for screen readers
The least common multiple (LCM) of 84 and 56 is $168$.
Steps to Solve
- 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: $$ 56 = 2^3 \times 7^1 $$
- 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)
- 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 $24 \times 7 = 168$.
- Final Result
The least common multiple of 84 and 56 is: $$ LCM(84, 56) = 168 $$
The least common multiple (LCM) of 84 and 56 is $168$.
More Information
The least common multiple is the smallest number that both 84 and 56 can divide without leaving a remainder. LCM is useful in various mathematical problems including finding common periods in real-life situations, such as scheduling.
Tips
- Confusing LCM with GCD. Always remember LCM is about finding the least common multiple while GCD is the greatest common divisor.
- Incorrectly identifying the prime factors. Ensure accurate factorization to avoid mistakes.