least common multiple of 27 and 63
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 27 and 63. To solve this, we will find the multiples of both numbers and then identify the smallest multiple that they share.
Answer
The least common multiple (LCM) of 27 and 63 is $189$.
Answer for screen readers
The least common multiple (LCM) of 27 and 63 is $189$.
Steps to Solve
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Factor the Numbers First, we will find the prime factorization of both numbers. For $27$, we can factor it as: $$ 27 = 3^3 $$ For $63$, we can factor it as: $$ 63 = 3^2 \times 7^1 $$
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Identify the Highest Powers Next, we will identify the highest power of each prime factor from both factorizations. The prime factors are $3$ and $7$.
- The highest power of $3$ is $3^3$ (from $27$).
- The highest power of $7$ is $7^1$ (from $63$).
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Multiply the Highest Powers Now, we multiply the highest powers to find the LCM. Thus, the LCM can be found using the formula: $$ \text{LCM} = 3^3 \times 7^1 $$
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Calculate the LCM Now we will calculate the LCM: $$ \text{LCM} = 27 \times 7 = 189 $$
The least common multiple (LCM) of 27 and 63 is $189$.
More Information
The LCM is the smallest number that is a multiple of both 27 and 63. Finding the LCM using prime factorization can also help understand how numbers can relate to one another in terms of their factors.
Tips
- Forgetting to include all prime factors when finding the LCM.
- Confusing the LCM with the greatest common divisor (GCD), which is the largest common factor instead of the smallest common multiple.
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