The correct answer and explanation is:
The correct answer is D. 480.
To find the Least Common Multiple (LCM) of 80, 96, and 120, we need to find the smallest positive integer that is evenly divisible by all three of these numbers. One of the most effective methods for this is using prime factorization.
First, we break down each number into its prime factors.
- For 80: 80 can be factored into 8 × 10. This further breaks down into (2 × 2 × 2) × (2 × 5), which in exponential form is 2⁴ × 5.
- For 96: 96 can be factored into 8 × 12. This further breaks down into (2 × 2 × 2) × (3 × 2 × 2), which in exponential form is 2⁵ × 3.
- For 120: 120 can be factored into 12 × 10. This further breaks down into (3 × 2 × 2) × (2 × 5), which in exponential form is 2³ × 3 × 5.
After finding the prime factorization for each number, we identify all the unique prime bases present, which are 2, 3, and 5. To calculate the LCM, we take the highest power of each of these prime bases from our factorizations.
- The highest power of 2 is 2⁵ (from the factorization of 96).
- The highest power of 3 is 3¹ (from the factorizations of 96 and 120).
- The highest power of 5 is 5¹ (from the factorizations of 80 and 120).
Next, we multiply these highest powers together to find the LCM:
LCM = 2⁵ × 3¹ × 5¹
LCM = 32 × 3 × 5
LCM = 96 × 5
LCM = 480
Another way to solve this, given the multiple-choice format, is to check the options starting from the smallest. The correct answer must be divisible by 80, 96, and 120.
Let’s test option D, 480:
- 480 ÷ 80 = 6
- 480 ÷ 96 = 5
- 480 ÷ 120 = 4
Since 480 is divisible by all three numbers and it is the smallest value among the options, it is the correct Least Common Multiple.
