What is the least common multiple of 36 and 42?
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 36 and 42. To solve this, we can use the prime factorization method or the method of listing multiples.
Answer
The LCM of 36 and 42 is $252$.
Answer for screen readers
The least common multiple (LCM) of 36 and 42 is 252.
Steps to Solve
- Find the Prime Factorization of 36
Start by breaking down 36 into its prime factors. $$ 36 = 6 \times 6 = 2 \times 3 \times 2 \times 3 = 2^2 \times 3^2 $$
- Find the Prime Factorization of 42
Next, do the same for 42. $$ 42 = 6 \times 7 = 2 \times 3 \times 7 = 2^1 \times 3^1 \times 7^1 $$
- Identify the Highest Powers of Each Prime Factor
Now, take the highest power of each prime factor from both numbers.
- For $2$: highest power is $2^2$
- For $3$: highest power is $3^2$
- For $7$: highest power is $7^1$
- Calculate the LCM
Multiply together the highest powers of each prime factor to find the LCM. $$ \text{LCM} = 2^2 \times 3^2 \times 7^1 $$
- Perform the Multiplication
Now, calculate the product: $$ \text{LCM} = 4 \times 9 \times 7 $$
Multiply the numbers step by step: $$ 4 \times 9 = 36 $$ $$ 36 \times 7 = 252 $$
The least common multiple (LCM) of 36 and 42 is 252.
More Information
The LCM is useful in many areas, such as finding common denominators in fractions or solving problems involving repeating events. It is the smallest positive multiple that is divisible by both numbers.
Tips
- Forgetting to multiply all the highest powers of the prime factors can lead to a wrong LCM.
- Mixing up the prime factors of each number can also cause errors. Always double-check the prime factorization step.
