Find the least common multiple (LCM) for each of the following groups of numbers
12, 20, and 35
That would be the number that contains the prime factors of all of the three numbers, for the maximum number of times that they appear in any single number.
12 = 3*2*2
20 = 2*2*5
35 = 5*7
The LCM is 2*2*3*5*7= 420
To find the least common multiple (LCM) for a group of numbers, you can follow these steps:
1. Prime Factorization: Begin by finding the prime factorization of each number in the group. Prime factorization breaks down a number into a product of prime numbers.
Let's find the prime factorization for each number:
- 12: The prime factors of 12 are 2 and 3. So, its prime factorization is 2^2 x 3^1.
- 20: The prime factors of 20 are 2 and 5. So, its prime factorization is 2^2 x 5^1.
- 35: The prime factors of 35 are 5 and 7. So, its prime factorization is 5^1 x 7^1.
2. Identify the Highest Power: Look at the prime factorization of each number and identify the highest power of each prime factor.
From the prime factorization, we can see that the highest power of:
- 2 is 2^2.
- 3 is 3^1.
- 5 is 5^1.
- 7 is 7^1.
3. Calculate the LCM: To find the LCM, multiply all the highest powers identified in step 2 together.
LCM = 2^2 x 3^1 x 5^1 x 7^1
Simplifying, we get:
LCM = 4 x 3 x 5 x 7
LCM = 420
Therefore, the least common multiple (LCM) for the numbers 12, 20, and 35 is 420.