Answer :
To find the least common multiple (LCM) of 70, 60, and 50, we need to determine the smallest number that is a multiple of all three numbers. Here is how you can find the LCM step by step:
1. Find the Prime Factorization of Each Number:
- 70: Prime factors are [tex]\(2 \times 5 \times 7\)[/tex].
- 60: Prime factors are [tex]\(2^2 \times 3 \times 5\)[/tex].
- 50: Prime factors are [tex]\(2 \times 5^2\)[/tex].
2. Determine the Highest Power of Each Prime Number Present:
- For 2, the highest power present is [tex]\(2^2\)[/tex] from 60.
- For 3, the highest power present is [tex]\(3^1\)[/tex] from 60.
- For 5, the highest power present is [tex]\(5^2\)[/tex] from 50.
- For 7, the highest power present is [tex]\(7^1\)[/tex] from 70.
3. Calculate the LCM by Multiplying the Highest Powers Together:
- LCM = [tex]\(2^2 \times 3^1 \times 5^2 \times 7^1\)[/tex].
4. Perform the Multiplication:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- Multiply these together: [tex]\(4 \times 3 = 12\)[/tex]
- Now multiply that result by 25: [tex]\(12 \times 25 = 300\)[/tex]
- Finally, multiply by 7: [tex]\(300 \times 7 = 2100\)[/tex]
Therefore, the least common multiple of 70, 60, and 50 is [tex]\(2100\)[/tex].
This matches with option J, so the correct answer is J. 2,100.
1. Find the Prime Factorization of Each Number:
- 70: Prime factors are [tex]\(2 \times 5 \times 7\)[/tex].
- 60: Prime factors are [tex]\(2^2 \times 3 \times 5\)[/tex].
- 50: Prime factors are [tex]\(2 \times 5^2\)[/tex].
2. Determine the Highest Power of Each Prime Number Present:
- For 2, the highest power present is [tex]\(2^2\)[/tex] from 60.
- For 3, the highest power present is [tex]\(3^1\)[/tex] from 60.
- For 5, the highest power present is [tex]\(5^2\)[/tex] from 50.
- For 7, the highest power present is [tex]\(7^1\)[/tex] from 70.
3. Calculate the LCM by Multiplying the Highest Powers Together:
- LCM = [tex]\(2^2 \times 3^1 \times 5^2 \times 7^1\)[/tex].
4. Perform the Multiplication:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- Multiply these together: [tex]\(4 \times 3 = 12\)[/tex]
- Now multiply that result by 25: [tex]\(12 \times 25 = 300\)[/tex]
- Finally, multiply by 7: [tex]\(300 \times 7 = 2100\)[/tex]
Therefore, the least common multiple of 70, 60, and 50 is [tex]\(2100\)[/tex].
This matches with option J, so the correct answer is J. 2,100.
