lcm of 24 and 21
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 24 and 21. To solve this, we can find the prime factorizations of both numbers and then determine the LCM based on those factors.
Answer
The least common multiple of \( 24 \) and \( 21 \) is \( 168 \).
Answer for screen readers
The least common multiple (LCM) of 24 and 21 is ( LCM = 168 ).
Steps to Solve
- Find the prime factorization of 24
The prime factors of 24 can be found by dividing it by the smallest prime numbers. $$ 24 = 2 \times 12 $$ $$ 12 = 2 \times 6 $$ $$ 6 = 2 \times 3 $$ Thus, the prime factorization of 24 is: $$ 24 = 2^3 \times 3^1 $$
- Find the prime factorization of 21
Similarly, we find the prime factors of 21. $$ 21 = 3 \times 7 $$ So, the prime factorization of 21 is: $$ 21 = 3^1 \times 7^1 $$
- Determine the LCM using the prime factorizations
To find the LCM, we take the highest power of each prime factor from both factorizations. From 24:
- $2^3$
- $3^1$
From 21:
- $3^1$
- $7^1$
We take:
- $2^3$ from 24
- $3^1$ from either (both share it)
- $7^1$ from 21
Therefore, the LCM is: $$ LCM = 2^3 \times 3^1 \times 7^1 $$
- Calculate the LCM
Now we can calculate the value: $$ LCM = 8 \times 3 \times 7 = 24 \times 7 = 168 $$
The least common multiple (LCM) of 24 and 21 is ( LCM = 168 ).
More Information
The least common multiple is a helpful concept in various mathematical applications, such as finding common denominators in fractions. The LCM is the smallest multiple that is common to both numbers, making it useful for solving problems involving ratios or periodic events.
Tips
- Failing to identify all prime factors correctly can lead to an incorrect LCM.
- Forgetting to take the highest power of each prime when calculating the LCM.
- Mixing the primes from each factorization instead of keeping them distinct.