What is the least common multiple of 12 and 28?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 12 and 28. To find the LCM, we can use the prime factorization method or the listing multiples method to determine the smallest multiple that both numbers share.
Answer
The least common multiple (LCM) of 12 and 28 is $84$.
Answer for screen readers
The least common multiple (LCM) of 12 and 28 is $84$.
Steps to Solve
- Prime Factorization of 12
We first find the prime factors of 12.
12 can be factored into primes as follows:
$$ 12 = 2^2 \times 3^1 $$
- Prime Factorization of 28
Next, we find the prime factors of 28.
28 can be factored into primes as follows:
$$ 28 = 2^2 \times 7^1 $$
- Identify Unique Prime Factors
Identify the unique prime factors from both sets:
- From 12: $2$ and $3$
- From 28: $2$ and $7$
The unique prime factors are $2$, $3$, and $7$.
- Take Highest Powers of Each Prime
Now, we take the highest power of each prime factor:
- For $2$: The highest power is $2^2$.
- For $3$: The highest power is $3^1$.
- For $7$: The highest power is $7^1$.
- Calculate the LCM
To find the LCM, multiply the highest powers of each prime factor:
$$ \text{LCM} = 2^2 \times 3^1 \times 7^1 $$
Calculating this gives:
$$ \text{LCM} = 4 \times 3 \times 7 = 84 $$
The least common multiple (LCM) of 12 and 28 is $84$.
More Information
The LCM is the smallest number into which both original numbers can be divided without any remainder. Finding the LCM using prime factorization is efficient, especially for larger numbers.
Tips
- Forgetting to include all unique prime factors: Ensure that you capture all prime factors from both numbers.
- Miscalculating the highest powers: Double-check that you take the maximum exponent for each prime number.
