College

What is the least common multiple of 70, 60, and [tex]$50$[/tex]?

F. [tex]60[/tex]
G. [tex]180[/tex]
H. [tex]210[/tex]
J. [tex]2,100[/tex]
K. [tex]210,000[/tex]

Answer :

To find the least common multiple (LCM) of the numbers 70, 60, and 50, we can follow these steps:

1. Find the Prime Factorization of Each Number:

- 70:
- 70 is divisible by 2: [tex]\( 70 \div 2 = 35 \)[/tex]
- 35 is divisible by 5: [tex]\( 35 \div 5 = 7 \)[/tex]
- 7 is a prime number
- Prime factors: [tex]\( 2 \times 5 \times 7 \)[/tex]

- 60:
- 60 is divisible by 2: [tex]\( 60 \div 2 = 30 \)[/tex]
- 30 is divisible by 2: [tex]\( 30 \div 2 = 15 \)[/tex]
- 15 is divisible by 3: [tex]\( 15 \div 3 = 5 \)[/tex]
- 5 is a prime number
- Prime factors: [tex]\( 2^2 \times 3 \times 5 \)[/tex]

- 50:
- 50 is divisible by 2: [tex]\( 50 \div 2 = 25 \)[/tex]
- 25 is divisible by 5: [tex]\( 25 \div 5 = 5 \)[/tex]
- 5 is a prime number
- Prime factors: [tex]\( 2 \times 5^2 \)[/tex]

2. Determine the Highest Power of Each Prime Factor Present:

- 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]\(3^1\)[/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]\(7^1\)[/tex] (from 70).

3. Multiply These Highest Powers to Get the LCM:

[tex]\[
\text{LCM} = 2^2 \times 3^1 \times 5^2 \times 7^1
\][/tex]

Calculate this:

[tex]\[
= 4 \times 3 \times 25 \times 7
\][/tex]

- [tex]\( 4 \times 3 = 12 \)[/tex]
- [tex]\( 12 \times 25 = 300 \)[/tex]
- [tex]\( 300 \times 7 = 2100 \)[/tex]

So, the least common multiple of 70, 60, and 50 is [tex]\( \boxed{2100} \)[/tex].