Answer :
The question asks us to find the Least Common Multiple (LCM) of several sets of numbers. The LCM of a set of numbers is the smallest number that is evenly divisible by each number in the set.
Here’s how to find the LCM step by step for each pair or set of numbers provided:
I. Find the LCM
a. 42, 63
Prime Factorization
- 42: [tex]2 \times 3 \times 7[/tex]
- 63: [tex]3^2 \times 7[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^1 \times 3^2 \times 7^1 = 126[/tex]
- So, the LCM is 126.
b. 60, 75
Prime Factorization
- 60: [tex]2^2 \times 3 \times 5[/tex]
- 75: [tex]3 \times 5^2[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^2 \times 3^1 \times 5^2 = 300[/tex]
- So, the LCM is 300.
c. 12, 18, 20
Prime Factorization
- 12: [tex]2^2 \times 3[/tex]
- 18: [tex]2 \times 3^2[/tex]
- 20: [tex]2^2 \times 5[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^2 \times 3^2 \times 5 = 180[/tex]
- So, the LCM is 180.
d. 36, 60, 72
Prime Factorization
- 36: [tex]2^2 \times 3^2[/tex]
- 60: [tex]2^2 \times 3 \times 5[/tex]
- 72: [tex]2^3 \times 3^2[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^3 \times 3^2 \times 5 = 360[/tex]
- So, the LCM is 360.
e. 144, 180, 384
Prime Factorization
- 144: [tex]2^4 \times 3^2[/tex]
- 180: [tex]2^2 \times 3^2 \times 5[/tex]
- 384: [tex]2^7 \times 3[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^7 \times 3^2 \times 5 = 5760[/tex]
- So, the LCM is 5760.
f. 48, 64, 72, 96, 108
Prime Factorization
- 48: [tex]2^4 \times 3[/tex]
- 64: [tex]2^6[/tex]
- 72: [tex]2^3 \times 3^2[/tex]
- 96: [tex]2^5 \times 3[/tex]
- 108: [tex]2^2 \times 3^3[/tex]
LCM Calculation
- Take the highest power of each prime number: [tex]2^6 \times 3^3 = 3456[/tex]
- So, the LCM is 3456.
In conclusion, the LCM of each set has been found using prime factorization, ensuring that we consider the highest power of each prime factor available in the numbers.
