To determine which number is a multiple of both 6 and 8, we need to understand the criteria for a number to be a multiple of two given numbers.
A number is a multiple of another number if it can be divided by that number without leaving a remainder. Specifically, if we are looking for a number that is a multiple of both 6 and 8, it must be divisible by their least common multiple (LCM).
Step-by-step Solution:
1. Compute the Least Common Multiple (LCM) of 6 and 8:
- The prime factorization of 6 is:
[tex]\( 6 = 2 \times 3 \)[/tex]
- The prime factorization of 8 is:
[tex]\( 8 = 2^3 \)[/tex]
- The LCM is found by taking the highest power of each prime number that appears in the factorizations:
[tex]\[ LCM(6, 8) = 2^3 \times 3 = 8 \times 3 = 24 \][/tex]
2. Check which of the given numbers is divisible by the LCM (24):
- Test [tex]\( 24 \)[/tex]:
[tex]\[ 24 \div 24 = 1 \quad \text{(no remainder, so 24 is a multiple of 24)} \][/tex]
- Test [tex]\( 3 \)[/tex]:
[tex]\[ 3 \div 24 \neq \text{integer} \quad \text{(not divisible, so 3 is not a multiple of 24)} \][/tex]
- Test [tex]\( 18 \)[/tex]:
[tex]\[ 18 \div 24 \neq \text{integer} \quad \text{(not divisible, so 18 is not a multiple of 24)} \][/tex]
Since 24 is the only number among the given options that is divisible by both 6 and 8:
Answer: 24
