lcm of 36 and 42
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 36 and 42. This involves finding the smallest positive integer that is divisible by both of these numbers.
Answer
$252$
Answer for screen readers
The least common multiple of 36 and 42 is $252$.
Steps to Solve
- Find the prime factorization of each number
For 36, we can break it down into prime factors: $$ 36 = 2^2 \times 3^2 $$
For 42, the prime factorization is: $$ 42 = 2^1 \times 3^1 \times 7^1 $$
- Identify the highest power of each prime factor
Next, we will identify the highest power of each prime factor present in the factorizations:
- For the prime number 2, the highest power is $2^2$ (from 36).
- For the prime number 3, the highest power is $3^2$ (from 36).
- For the prime number 7, the highest power is $7^1$ (from 42).
- Multiply the highest powers of each prime factor
Now we will multiply these highest powers together to find the least common multiple (LCM): $$ LCM(36, 42) = 2^2 \times 3^2 \times 7^1 $$
- Calculate the product to find the LCM
Now let's calculate this product: $$ LCM(36, 42) = 4 \times 9 \times 7 $$
First, calculate $4 \times 9 = 36$. Then, calculate $36 \times 7 = 252$.
So, the LCM is: $$ LCM(36, 42) = 252 $$
The least common multiple of 36 and 42 is $252$.
More Information
The least common multiple (LCM) is useful in various applications, such as finding common denominators for fractions and scheduling events that occur at different intervals.
Tips
- Confusing LCM with GCD (Greatest Common Divisor). Remember, LCM is the smallest multiple, while GCD is the largest divisor.
- Failing to consider all prime factors in higher powers. Make sure to always take the maximum power of each prime factor present.
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