Least common multiple of 18 and 42

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 18 and 42. To find the LCM, we can use the prime factorization or the method of listing multiples. The LCM is the smallest multiple that is shared by both numbers.

Answer

The least common multiple (LCM) of 18 and 42 is $126$.

Steps to Solve

Find the Prime Factorization of Each Number

To find the LCM, we first need to determine the prime factorization of both numbers.

For 18, the prime factorization is: $$ 18 = 2 \times 3^2 $$
For 42, the prime factorization is: $$ 42 = 2 \times 3 \times 7 $$

Identify the Highest Powers of Each Prime Factor

Next, we take the highest power of each prime factor from both factorizations to find the LCM.

From 18:
- For 2: $2^1$
- For 3: $3^2$
From 42:
- For 2: $2^1$
- For 3: $3^1$
- For 7: $7^1$

The highest powers are:

$2^1$
$3^2$
$7^1$

Multiply the Highest Powers Together

Now we multiply these highest powers to find the LCM:

$$ LCM = 2^1 \times 3^2 \times 7^1 $$

Calculate the multiplication step by step:

First, compute $3^2 = 9$.
Then, multiply: $$ LCM = 2 \times 9 \times 7 $$
$2 \times 9 = 18$
Finally, $18 \times 7 = 126$

Thus, we find the LCM of 18 and 42 is 126.

The least common multiple (LCM) of 18 and 42 is 126.

More Information

The LCM is useful in many mathematical operations, particularly when adding or subtracting fractions with different denominators. The method shown here using prime factorization ensures that you correctly account for all the prime factors involved.

Tips

Forgetting to include all prime factors when comparing their highest powers.
Confusing the concept of LCM with the greatest common divisor (GCD), which is the largest divisor.

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