Answer :

To find the least common multiple (LCM) of 70, 60, and 50, let's understand what LCM is first. The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. Here’s a step-by-step guide to finding the LCM:

1. Prime Factorization:
- First, we find the prime factorization of each number.
- 70: [tex]\(70 = 2 \times 5 \times 7\)[/tex]
- 60: [tex]\(60 = 2^2 \times 3 \times 5\)[/tex]
- 50: [tex]\(50 = 2 \times 5^2\)[/tex]

2. Identify the Highest Power for Each Prime:
- Look at each prime factor that appears in any of the numbers and take the highest power from all of them.
- For the prime number 2, the highest power is [tex]\(2^2\)[/tex] (from 60).
- For the prime number 3, the highest power is [tex]\(3\)[/tex] (from 60).
- For the prime number 5, the highest power is [tex]\(5^2\)[/tex] (from 50).
- For the prime number 7, the highest power is [tex]\(7\)[/tex] (from 70).

3. Calculate the LCM:
- Multiply these highest powers together to get the LCM:
- LCM = [tex]\(2^2 \times 3 \times 5^2 \times 7\)[/tex]

4. Perform the Multiplication:
- First, calculate [tex]\(2^2 = 4\)[/tex].
- Then, multiply [tex]\(4 \times 3 = 12\)[/tex].
- Next, [tex]\(12 \times 25 = 300\)[/tex] (since [tex]\(5^2 = 25\)[/tex]).
- Finally, [tex]\(300 \times 7 = 2100\)[/tex].

Therefore, the least common multiple of 70, 60, and 50 is [tex]\(\boxed{2,100}\)[/tex]. So, the correct answer is option J, 2,100.

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