least common multiple of 6 and 13

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 6 and 13. The LCM is the smallest number that is a multiple of both 6 and 13. To find it, we can list the multiples of each number or use the formula involving the greatest common divisor (GCD).

Answer

$78$

Steps to Solve

Identify the numbers
We need to find the least common multiple (LCM) of the numbers 6 and 13.
Find the multiples of each number
List some multiples of each number:
For 6:

$6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...$

For 13:

$13, 26, 39, 52, 65, 78, ...$

Find the common multiples
Identify the smallest number that appears in both lists. Since 6 and 13 do not share multiples until their product, let's skip to the next method.
Use the formula involving GCD
The formula to calculate the LCM using GCD is:
$$ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} $$
In our case:

$a = 6$
$b = 13$

Calculate the GCD of 6 and 13
Since 6 and 13 have no common factors other than 1, their GCD is:
$$ \text{GCD}(6, 13) = 1 $$
Calculate the LCM using GCD
Now we can find the LCM:
$$ \text{LCM}(6, 13) = \frac{|6 \times 13|}{1} = \frac{78}{1} = 78 $$

The least common multiple of 6 and 13 is $78$.

More Information

The least common multiple (LCM) can also be viewed through the concept of prime factorization. However, since 6 and 13 share no prime factors (6 = $2 \times 3$ and 13 is a prime number), the LCM can be calculated efficiently using the GCD formula.

Tips

Mixing up GCD and LCM. Remember that GCD finds the greatest common factor, while LCM finds the least common multiple.
Failing to recognize that for two relatively prime numbers (like 6 and 13), the LCM is simply their product.

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