Find the Least Common Multiple of 12, 30, and 32
The Correct Answer and Explanation is:
The least common multiple (LCM) of 12, 30, and 32 is 480.
Explanation:
The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the given numbers. To determine the LCM of 12, 30, and 32, we follow a systematic approach using prime factorization.
Step 1: Prime Factorization
We first break each number down into its prime factors:
- 12 = 22×32^2 \times 3
- 30 = 2×3×52 \times 3 \times 5
- 32 = 252^5
Step 2: Identifying the Highest Powers of Each Prime Factor
To find the LCM, we take the highest powers of all prime factors present:
- The highest power of 2 is 252^5 (from 32).
- The highest power of 3 is 313^1 (from 12 and 30).
- The highest power of 5 is 515^1 (from 30).
Thus, the LCM is found by multiplying the highest powers:
LCM=25×31×51LCM = 2^5 \times 3^1 \times 5^1
LCM=32×3×5=480LCM = 32 \times 3 \times 5 = 480
Step 3: Confirming Multiples
Now, we check that 480 is divisible by 12, 30, and 32:
- 480÷12=40480 \div 12 = 40 ✅
- 480÷30=16480 \div 30 = 16 ✅
- 480÷32=15480 \div 32 = 15 ✅
Since 480 is a multiple of all three numbers, it is the correct Least Common Multiple (LCM).
This method ensures accuracy and is widely used in books and textbooks to find the LCM of multiple numbers efficiently.
