Find the LCM of the list of numbers. 9 and 21 The LCM of 9 and 21 is
The Correct Answer and Explanation is:
To find the Least Common Multiple (LCM) of 9 and 21, we will follow these steps:
Step 1: Prime Factorization
We start by finding the prime factorization of each number.
- Prime factorization of 9:
9 = 3 × 3 = 3² - Prime factorization of 21:
21 = 3 × 7
Step 2: Identify the Highest Powers of All Prime Factors
Now, we list the prime factors of both numbers and select the highest powers of each prime.
- For the prime 3, the highest power is 3² (from 9).
- For the prime 7, the highest power is 7 (from 21).
Step 3: Multiply the Highest Powers
To find the LCM, we multiply these highest powers:LCM=32×7=9×7=63LCM = 3² × 7 = 9 × 7 = 63LCM=32×7=9×7=63
Step 4: Verify the Result
To verify, we check if 63 is divisible by both 9 and 21:
- 63 ÷ 9 = 7 (an integer, so 63 is divisible by 9).
- 63 ÷ 21 = 3 (an integer, so 63 is divisible by 21).
Since 63 is divisible by both 9 and 21, we confirm that it is indeed the LCM.
Conclusion:
The LCM of 9 and 21 is 63.
Explanation:
The LCM is the smallest number that both 9 and 21 divide into without leaving a remainder. We use the prime factorization method to find it. First, we find the prime factors of each number, then we take the highest powers of all primes involved. The LCM is the product of these highest powers. In this case, 3² and 7 are the highest powers, giving us an LCM of 63. This is the smallest multiple of both 9 and 21.
