lcm of 42 and 24

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 42 and 24. To find the LCM, we will identify the prime factors of both numbers and then determine the smallest multiple that is common to both.

Answer

The LCM of 42 and 24 is $168$.

Steps to Solve

Find prime factors of 42
To find the least common multiple (LCM), we first need to determine the prime factors of 42.
The prime factorization of 42 is:
$$ 42 = 2 \times 3 \times 7 $$
Find prime factors of 24
Next, we find the prime factors of 24.
The prime factorization of 24 is:
$$ 24 = 2^3 \times 3 $$
List unique prime factors
Now, we list all unique prime factors from both numbers:

From 42: 2, 3, 7
From 24: 2, 3

The unique prime factors are: 2, 3, and 7.

Determine the highest power of each prime factor
We need to take the highest power of each prime factor:

For 2, the highest power is $2^3$ (from 24)
For 3, the highest power is $3^1$ (from both 42 and 24)
For 7, the highest power is $7^1$ (from 42)

Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
$$ LCM = 2^3 \times 3^1 \times 7^1 $$
Calculating this gives:
$$ LCM = 8 \times 3 \times 7 $$
Final calculation
Continuing the calculation:
$$ 8 \times 3 = 24 $$
Then,
$$ 24 \times 7 = 168 $$

So, the least common multiple of 42 and 24 is 168.

The least common multiple (LCM) of 42 and 24 is $168$.

More Information

The least common multiple (LCM) is useful in various applications, such as finding common denominators in fraction addition. An interesting fact is that the LCM can also be computed using the relationship between LCM and greatest common divisor (GCD):
$$ LCM(a, b) = \frac{a \times b}{GCD(a, b)} $$

Tips

Forgetting to consider the highest powers of the prime factors can lead to an incorrect LCM.
Miscalculating the products when multiplying the prime factors together.

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