LCM of 21 and 49
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 21 and 49. To find the LCM, we will identify the multiples of both numbers and determine the smallest multiple they have in common.
Answer
The least common multiple of 21 and 49 is $147$.
Answer for screen readers
The least common multiple (LCM) of 21 and 49 is $147$.
Steps to Solve
- Find the prime factorization of each number
To begin, we will factor both numbers into their prime components.
For 21:
$21 = 3 \times 7$
For 49:
$49 = 7 \times 7 = 7^2$
- List all unique prime factors
Next, we will list all the unique prime factors identified from both factorizations. We have:
- From 21: 3, 7
- From 49: 7
So, the unique prime factors are 3 and 7.
- Determine the highest powers of each prime factor
Now, we will take the highest power of each unique prime factor.
- For 3: The highest power is $3^1$ (from 21)
- For 7: The highest power is $7^2$ (from 49)
- Multiply the highest powers together
Finally, we will multiply the highest powers of the unique prime factors to find the LCM.
$$LCM = 3^1 \times 7^2$$
Calculating this gives:
$$LCM = 3 \times 49 = 147$$
The least common multiple (LCM) of 21 and 49 is $147$.
More Information
The least common multiple is useful in various mathematical contexts, including adding fractions with different denominators or finding common time intervals in problems involving periods or cycles.
Tips
- Not using the highest power: A common mistake is to take the product of the prime factors instead of their highest powers. Always check that you include the highest exponent when finding the LCM.
- Forgetting about unique factors: Ensure you consider all unique prime factors from both numbers.
