What is the lowest common multiple of 18 and 42?
Understand the Problem
The question is asking for the lowest common multiple (LCM) of the numbers 18 and 42, which involves finding the smallest positive integer that is divisible by both numbers.
Answer
The lowest common multiple of 18 and 42 is \(126\).
Answer for screen readers
The lowest common multiple of 18 and 42 is (126).
Steps to Solve
- Find the Prime Factorization of Each Number
Start by determining the prime factors of both 18 and 42.
For 18:
$$ 18 = 2 \times 3^2 $$
For 42:
$$ 42 = 2 \times 3 \times 7 $$
- Identify the Highest Powers of Each Prime Factor
Next, list the unique prime factors and their highest powers found in the factorizations.
- The prime factors are: 2, 3, and 7.
- The highest powers of each are:
- For 2: $2^1$
- For 3: $3^2$
- For 7: $7^1$
- Calculate the LCM Using the Highest Powers
Multiply the highest powers of all prime factors to find the LCM.
$$ LCM = 2^1 \times 3^2 \times 7^1 $$
This simplifies to:
$$ LCM = 2 \times 9 \times 7 $$
- Perform the Final Calculation
Now, calculate the product step-by-step:
First, calculate $2 \times 9$:
$$ 2 \times 9 = 18 $$
Then multiply by 7:
$$ 18 \times 7 = 126 $$
Thus, the lowest common multiple (LCM) of 18 and 42 is 126.
The lowest common multiple of 18 and 42 is (126).
More Information
The lowest common multiple (LCM) is useful in various applications, such as finding common denominators in fractions and solving problems involving periodic events. Knowing how to find the LCM can help simplify many math problems.
Tips
- Confusing LCM with GCD (Greatest Common Divisor). Remember that LCM gives the smallest common multiple, while GCD gives the largest common divisor.
- Miscalculating the prime factorization, which can lead to an incorrect LCM.
