Final answer:
The least common multiple of 12, 18, and 30 is calculated using prime factorization, with the highest powers of all prime factors being taken into account.
Step-by-step explanation:
The least common multiple (LCM) of 12, 18, and 30 needs to be found to determine the smallest number that is a multiple of all three. To find the LCM, we can use the prime factorization method:
- 12 = 2² × 3
- 18 = 2 × 3²
- 30 = 2 × 3 × 5
Now, we take the highest powers of all prime factors that appear in any of the numbers:
LCM = 2² (which is 4) × 3² (which is 9) × 5 = 4 × 9 × 5 = 36 × 5 = 180.
The least common multiple of 12, 18, and 30 is 180. This number is the smallest number that all three numbers can divide into without leaving a remainder.