lcm of 42 and 14

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 42 and 14. To find the LCM, we need to determine the smallest number that is a multiple of both 42 and 14.

Answer

The LCM of 42 and 14 is $42$.

Steps to Solve

Prime Factorization of the Numbers

First, we will find the prime factorization of both numbers.

For 42:

$42 = 2 \times 3 \times 7$

For 14:

$14 = 2 \times 7$

Identify the Highest Power of Each Prime Factor

Now we identify the highest power of each prime factor from both numbers.

From 42:
- 2 appears as $2^1$
- 3 appears as $3^1$
- 7 appears as $7^1$
From 14:
- 2 appears as $2^1$
- 7 appears as $7^1$

The highest powers are:

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

Calculate the LCM Using the Highest Powers

Now, we multiply the highest powers of all prime factors together to find the LCM.

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

Perform the Multiplication

Calculate the multiplication:

[ \text{LCM} = 2 \times 3 \times 7 = 6 \times 7 = 42 ]

The least common multiple of 42 and 14 is $42$.

More Information

The least common multiple (LCM) is useful in solving problems involving fractions, adding or subtracting fractions with different denominators, and finding common cycles in events, like scheduling.

Tips

Forgetting to take the highest power of each prime factor when determining the LCM.
Confusing the LCM with the greatest common divisor (GCD). The LCM is the smallest common multiple, while the GCD is the largest common factor.

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