Find the LCM of each number using prime factorization: 35, 75, 50.
The Correct Answer and Explanation is:
To find the Least Common Multiple (LCM) of the numbers 35, 75, and 50 using prime factorization, follow these steps:
Step 1: Prime Factorization of Each Number
- 35:
35 is divisible by 5 (because it ends in 5).
35 ÷ 5 = 7.
So, 35 = 5 × 7. - 75:
75 is divisible by 5 (since it ends in 5).
75 ÷ 5 = 15.
15 is also divisible by 5.
15 ÷ 5 = 3.
So, 75 = 5² × 3. - 50:
50 is divisible by 5.
50 ÷ 5 = 10.
10 is divisible by 5 again.
10 ÷ 5 = 2.
So, 50 = 5² × 2.
Step 2: Identify the Highest Powers of Each Prime Factor
Now, identify the highest powers of each prime factor present in the factorizations:
- Prime factor 2: The highest power is 2¹ (from 50).
- Prime factor 3: The highest power is 3¹ (from 75).
- Prime factor 5: The highest power is 5² (from both 75 and 50).
- Prime factor 7: The highest power is 7¹ (from 35).
Step 3: Multiply the Highest Powers Together
To find the LCM, multiply the highest powers of each prime factor:LCM=21×31×52×71LCM = 2¹ × 3¹ × 5² × 7¹LCM=21×31×52×71LCM=2×3×25×7LCM = 2 × 3 × 25 × 7LCM=2×3×25×7LCM=2×3=6LCM = 2 × 3 = 6LCM=2×3=66×25=1506 × 25 = 1506×25=150150×7=1050150 × 7 = 1050150×7=1050
Final Answer:
The LCM of 35, 75, and 50 is 1050.
Explanation:
- The LCM is the smallest number that is divisible by each of the given numbers.
- Using prime factorization, you find the highest powers of each prime factor from all the numbers and multiply them together.
- By following this method, you ensure that the LCM will include all factors of each original number without any unnecessary repeats.
