LCM of 18 and 42
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 18 and 42. To solve this, we can find the multiples of both numbers and determine the smallest common multiple.
Answer
The least common multiple of 18 and 42 is $126$.
Answer for screen readers
The least common multiple of 18 and 42 is 126.
Steps to Solve
- Find the Prime Factorization of Each Number
First, we find the prime factors of both 18 and 42.
- The prime factorization of $18$ is $2 \times 3^2$.
- The prime factorization of $42$ is $2 \times 3 \times 7$.
- Identify the Highest Powers of Each Prime
Next, we identify the highest powers of all prime factors involved.
- For the prime $2$: Highest power is $2^1$ (from both).
- For the prime $3$: Highest power is $3^2$ (from 18).
- For the prime $7$: Highest power is $7^1$ (from 42).
- Multiply the Highest Powers Together
Now, we multiply these highest powers to find the LCM. $$ LCM = 2^1 \times 3^2 \times 7^1 $$
Calculating this gives: $$ LCM = 2 \times 9 \times 7 $$
- Perform the Multiplication
First multiply $2$ and $9$: $$ 2 \times 9 = 18 $$
Then multiply by $7$: $$ 18 \times 7 = 126 $$
Thus, the least common multiple of 18 and 42 is $126$.
The least common multiple of 18 and 42 is 126.
More Information
The least common multiple (LCM) is the smallest positive integer that is divisible by both numbers. It is useful in various mathematical applications, particularly in adding or subtracting fractions.
Tips
- Not finding the full prime factorization: Make sure to completely factor numbers into primes before identifying the LCM.
- Incorrectly assuming the LCM is simply the product of the two numbers: The LCM is based on the highest powers of prime factors, not just multiplication.
