lcm of 27 and 63
Understand the Problem
The question is asking to calculate the least common multiple (LCM) of the numbers 27 and 63. We can solve this by finding the prime factorization of both numbers and using those factors to determine the LCM.
Answer
The 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
- Prime Factorization of 27
To find the prime factorization of 27, we divide it by prime numbers.
27 can be divided by 3:
$$ 27 \div 3 = 9 $$
9 can again be divided by 3:
$$ 9 \div 3 = 3 $$
Finally, 3 can be divided by 3:
$$ 3 \div 3 = 1 $$
So, the prime factorization of 27 is:
$$ 27 = 3^3 $$
- Prime Factorization of 63
Next, we do the same for 63:
63 can be divided by 3:
$$ 63 \div 3 = 21 $$
21 can again be divided by 3:
$$ 21 \div 3 = 7 $$
7 is itself a prime number, so we stop here.
Thus, the prime factorization of 63 is:
$$ 63 = 3^2 \times 7^1 $$
- Determining the LCM
To find the LCM, we take the highest power of each prime factor from the factorizations of both numbers.
- For the prime number 3, the highest power is $3^3$.
- For the prime number 7, the highest power is $7^1$.
So we compute the LCM as follows:
$$ LCM = 3^3 \times 7^1 $$
- Calculate the LCM
Now we calculate the LCM:
First calculate $3^3$:
$$ 3^3 = 27 $$
Then multiply by $7$:
$$ LCM = 27 \times 7 = 189 $$
The least common multiple (LCM) of 27 and 63 is $189$.
More Information
The least common multiple is useful when working with fractions or when you want to find a common multiple of two numbers for adding or subtracting. The LCM is the smallest number that is a multiple of both original numbers.
Tips
- Confusing LCM with greatest common divisor (GCD), which is the largest number that divides both.
- Forgetting to take the highest power of each prime factor when calculating the LCM.