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].
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].
