Find the lowest common multiple of $24$, $36$ and $40$.

### Concept & Strategy The Lowest Common Multiple (LCM) of a set of numbers is the smallest positive integer that is uniformly divisible by all the numbers in the set. The most structured way to find the LCM of multiple integers is to use **Prime Factorization**. ### Step-by-Step Solution * **Step 1: Prime factorize each number.** * $24 = 8 \times 3 = 2^3 \times 3^1$ * $36 = 4 \times 9 = 2^2 \times 3^2$ * $40 = 8 \times 5 = 2^3 \times 5^1$ * **Step 2: Apply the LCM Rule.** The LCM is found by taking the highest power of each unique prime factor present across all the numbers. * Highest power of $2$ is $2^3$ * Highest power of $3$ is $3^2$ * Highest power of $5$ is $5^1$ * **Step 3: Calculate the final value.** $$LCM = 2^3 \times 3^2 \times 5^1$$ $$LCM = 8 \times 9 \times 5$$ $$LCM = 72 \times 5$$ $$LCM = 360$$ ### Exam Strategy & Shortcut Use the **Option Elimination Strategy**. The LCM must be a multiple of $40$, meaning it must end in $0$ (all options do). It must be a multiple of $36$, which means it must be divisible by $9$ (sum of digits must be a multiple of $9$). * $120$: Sum = $3$ (Not divisible by 9) * $240$: Sum = $6$ (Not divisible by 9) * $360$: Sum = $9$ (Divisible by 9!) * $480$: Sum = $12$ (Not divisible by 9) Only $360$ satisfies the divisibility rule for $9$, making it the correct answer in seconds without full calculations. ### Common Pitfall Using the "ladder" or common division method can sometimes lead to arithmetic errors if you skip a prime factor. Furthermore, students often forget to check divisibility by $9$ when $36$ or $72$ are in the mix, missing out on a massive time-saving shortcut. ### Final Answer **Therefore, the correct answer is 360.**