lcm of 28 and 40

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 28 and 40. To solve this, we would typically find the multiples of both numbers and determine the smallest multiple they have in common, or use prime factorization to calculate the LCM.

Answer

$280$

Steps to Solve

Prime Factorization of 28 First, we break down the number 28 into its prime factors.

$$ 28 = 2^2 \times 7 $$

Prime Factorization of 40 Next, we do the same for the number 40.

$$ 40 = 2^3 \times 5 $$

Identify the Highest Powers of Each Prime We need to select the highest powers of all the prime factors involved.

For the prime number 2, the maximum power from both factorizations is $2^3$ (from 40).
For the prime number 5, the maximum power is $5^1$ (from 40).
For the prime number 7, the maximum power is $7^1$ (from 28).

Multiply the Highest Powers Together Now we multiply these highest powers together to find the LCM.

$$ \text{LCM} = 2^3 \times 5^1 \times 7^1 $$

Calculate the LCM Now we calculate the final value:

$$ \text{LCM} = 8 \times 5 \times 7 $$

Calculating this step-by-step gives us:

First calculate $8 \times 5 = 40$.
Then, $40 \times 7 = 280$.

So,

$$ \text{LCM}(28, 40) = 280 $$

The least common multiple (LCM) of 28 and 40 is $280$.

More Information

The least common multiple is useful in various applications where you need to find a common size or synchronizing events. For example, if two people are cycling at different speeds, the LCM can tell us how often they will meet at the starting point.

Tips

Ignoring the highest powers: A common mistake is forgetting to take the highest power of each prime factor. Always ensure you include the maximum power from the factorizations.
Miscalculating the multiplication: It’s easy to make arithmetic errors when multiplying the numbers together; double-check your calculations.

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