lcm of 42 and 18
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 42 and 18. To solve this, we will first find the prime factorization of both numbers and then use these factors to find the LCM.
Answer
The LCM of 42 and 18 is $126$.
Answer for screen readers
The least common multiple (LCM) of 42 and 18 is 126.
Steps to Solve
- Find the prime factorization of 42
To find the prime factors of 42, we divide it by the smallest prime number until we get 1.
$$ 42 \div 2 = 21 \ 21 \div 3 = 7 \ 7 \div 7 = 1 $$
So the prime factorization of 42 is $2^1 \times 3^1 \times 7^1$.
- Find the prime factorization of 18
Similarly, we find the prime factors of 18.
$$ 18 \div 2 = 9 \ 9 \div 3 = 3 \ 3 \div 3 = 1 $$
So the prime factorization of 18 is $2^1 \times 3^2$.
- Determine the LCM using prime factors
To find the LCM, we take the highest power of each prime number present in the factorizations.
- For the prime number 2: The highest power is $2^1$.
- For the prime number 3: The highest power is $3^2$.
- For the prime number 7: The highest power is $7^1$.
So, the LCM is:
$$ LCM = 2^1 \times 3^2 \times 7^1 $$
- Calculate the LCM
Now we calculate:
$$ LCM = 2 \times 9 \times 7 $$
Calculating step by step,
First, calculate $2 \times 9 = 18$.
Then multiply $18 \times 7 = 126$.
So, $LCM(42, 18) = 126$.
The least common multiple (LCM) of 42 and 18 is 126.
More Information
The LCM of two numbers is the smallest multiple that is evenly divisible by both numbers. Finding the LCM through prime factorization is a common method in basic number theory. This technique can also be applied to more than two numbers.
Tips
- Not taking the highest powers of prime factors. Ensure to count the highest power for each prime across both factorizations.
- Miscalculating the multiplication of the prime factors. Take your time to perform each multiplication step carefully.
