What is the LCM of 40 and 28?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 40 and 28, which is a mathematical concept used to find the smallest positive integer that is divisible by both numbers.
Answer
The least common multiple of 40 and 28 is $280$.
Answer for screen readers
The least common multiple of 40 and 28 is $280$.
Steps to Solve
- Prime Factorization of 40
First, we will find the prime factors of 40.
40 can be factored as: $$ 40 = 2^3 \times 5^1 $$
- Prime Factorization of 28
Next, we will find the prime factors of 28.
28 can be factored as: $$ 28 = 2^2 \times 7^1 $$
- Identify the Highest Powers
For the LCM, we take the highest powers of all prime factors present in both factorizations.
From the factorizations:
- For prime number 2: highest power is $2^3$ from 40.
- For prime number 5: highest power is $5^1$ from 40.
- For prime number 7: highest power is $7^1$ from 28.
- Multiply the Highest Powers Together
Now we multiply the highest powers together to find the LCM:
$$ \text{LCM} = 2^3 \times 5^1 \times 7^1 $$
Now we calculate this step-by-step:
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First, calculate $2^3$: $$ 2^3 = 8 $$
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Then multiply by $5$: $$ 8 \times 5 = 40 $$
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Finally, multiply by $7$: $$ 40 \times 7 = 280 $$
- Final LCM Value
Thus, the least common multiple of 40 and 28 is:
$$ \text{LCM}(40, 28) = 280 $$
The least common multiple of 40 and 28 is $280$.
More Information
The least common multiple (LCM) is useful in various mathematical problems, particularly when adding or subtracting fractions with different denominators. Finding the LCM allows us to express those fractions with a common denominator.
Tips
- Not taking the highest power of each prime factor. It's essential to consider the maximum power present for each prime when calculating the LCM.
- Forgetting to include all prime factors present in both numbers during multiplication.
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