Find the least number which is exactly divisible by $32$, $36$, $45$, $60$ and $80$.

### Concept & Formula When asked for the "least number exactly divisible" by a set of numbers, you need to find their Lowest Common Multiple (L.C.M.). $$ \text{Required Number} = \text{L.C.M.}(a, b, c, \dots) $$ ### Step-by-Step Solution * **Given:** The target divisors are $32$, $36$, $45$, $60$, and $80$. * Use the prime factorization or common division method. Let's list out prime factors: * $32 = 2^5$ * $36 = 2^2 \times 3^2$ * $45 = 3^2 \times 5^1$ * $60 = 2^2 \times 3^1 \times 5^1$ * $80 = 2^4 \times 5^1$ * To find the L.C.M., take the highest power of every prime factor present across all numbers: * Highest power of $2$ is $2^5$ (from $32$). * Highest power of $3$ is $3^2$ (from $36$ or $45$). * Highest power of $5$ is $5^1$ (from $45$, $60$, or $80$). * Multiply these maximum prime powers together: * $\text{L.C.M.} = 2^5 \times 3^2 \times 5^1$ * $\text{L.C.M.} = 32 \times 9 \times 5$ * $\text{L.C.M.} = 32 \times 45 = 1440$ ### Exam Strategy & Shortcut Use **Divisibility Rules on Options**. The correct number must be divisible by $36$ (so it must be divisible by $9$) and by $32$ (so its last $5$ digits must be divisible by $32$, or you can just test). Check options for divisibility by $9$ (sum of digits): $720$, $1080$, $1440$, $2880$ all work. Next, check divisibility by $32$: $720/32 = 22.5$ (Fail), $1080/32 = 33.75$ (Fail). $1440/32 = 45$ (Pass!). Thus, $1440$ is the least number that works. ### Common Pitfall Using the continuous division (ladder) method on so many large numbers can easily lead to a small arithmetic mistake in one column, ruining the whole L.C.M. Prime factorization is generally safer, or explicitly combining smaller L.C.M.s (e.g., L.C.M. of $32$ and $80$ is $160$, then compare with the rest). ### Final Answer **Therefore, the correct answer is 1440.**