What is the least common multiple of 28 and 20?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 28 and 20. To find the LCM, we will determine the smallest multiple that both numbers share.
Answer
$140$
Answer for screen readers
The least common multiple (LCM) of 28 and 20 is $140$.
Steps to Solve
- Find the prime factorization of each number
To find the LCM, we need to break down each number into its prime factors.
For 28: $$ 28 = 2^2 \times 7^1 $$
For 20: $$ 20 = 2^2 \times 5^1 $$
- Identify the highest power of each prime factor
Next, we will take each unique prime factor from both factorizations and determine the highest exponent used.
- For the prime factor 2: the highest power is $2^2$ (from both 28 and 20).
- For the prime factor 5: the highest power is $5^1$ (from 20).
- For the prime factor 7: the highest power is $7^1$ (from 28).
- Calculate the LCM using the highest powers
Now we multiply these highest powers together to find the LCM.
$$ LCM = 2^2 \times 5^1 \times 7^1 $$
Calculating this gives:
$$ LCM = 4 \times 5 \times 7 $$
- Final multiplication
Now we perform the multiplication step-by-step:
First, calculate $4 \times 5$: $$ 4 \times 5 = 20 $$
Then, multiply the result by 7: $$ 20 \times 7 = 140 $$
Thus, the least common multiple of 28 and 20 is 140.
The least common multiple (LCM) of 28 and 20 is $140$.
More Information
The least common multiple (LCM) is significant in various mathematical concepts, including solving problems that involve finding common denominators, scheduling repeat events, and more. The LCM helps to ensure that multiple quantities align perfectly.
Tips
- Forgetting to consider all unique prime factors.
- Using the lowest instead of the highest power of each prime factor.
- Incorrectly multiplying the prime factors together.