Find the greatest possible length which can be used to measure exactly the lengths $4\text{ m } 95\text{ cm}$, $9\text{ m}$ and $16\text{ m } 65\text{ cm}$.

### Concept & Logic When a question asks for the "greatest possible length," "maximum capacity," or "largest size" that can exactly measure or divide multiple other quantities, you are looking for the Highest Common Factor (H.C.F.). Crucially, to avoid errors with decimals or mixed units, you must convert all measurements to the smallest common unit (in this case, centimeters) before performing any calculations. ### Step-by-Step Solution **Step 1: Convert all lengths to centimeters.** * Recall that $1\text{ m} = 100\text{ cm}$. * Length 1: $4\text{ m } 95\text{ cm} = 400\text{ cm} + 95\text{ cm} = 495\text{ cm}$. * Length 2: $9\text{ m} = 900\text{ cm}$. * Length 3: $16\text{ m } 65\text{ cm} = 1600\text{ cm} + 65\text{ cm} = 1665\text{ cm}$. **Step 2: Find the H.C.F. of the converted lengths ($495$, $900$, $1665$).** * Let's use prime factorization for each number: $495 = 5 \times 99 = 3^2 \times 5 \times 11$ $900 = 9 \times 100 = 3^2 \times 2^2 \times 5^2$ $1665 = 5 \times 333 = 5 \times 9 \times 37 = 3^2 \times 5 \times 37$ **Step 3: Calculate the product of common prime factors.** * The common prime bases across all three factorizations are $3$ and $5$. * The lowest power of $3$ present is $3^2$. * The lowest power of $5$ present is $5^1$. * H.C.F. = $3^2 \times 5^1$ * H.C.F. = $9 \times 5 = 45$. The greatest possible measuring length is $45\text{ cm}$. ### Exam Strategy & Shortcut Use the **Options and Divisibility Rules** to skip full factorization. The lengths are $495$, $900$, and $1665$. Because they all end in $0$ or $5$, they are definitively divisible by $5$. Are they divisible by $9$? Check the sum of their digits: $4+9+5 = 18$ (Divisible by $9$) $9+0+0 = 9$ (Divisible by $9$) $1+6+6+5 = 18$ (Divisible by $9$) Since they are all divisible by both $5$ and $9$ (which are co-prime), they must be divisible by $45$. Look at the options: $45\text{ cm}$ is present. Select it immediately and move on. ### Common Pitfall A classic mistake is attempting to find the H.C.F. while the numbers are still in mixed units (meters and centimeters) or decimal format ($4.95$, $9.00$, $16.65$), which leads to massive alignment errors. Always normalize to whole numbers before starting. ### Final Answer **Therefore, the correct answer is $45\text{ cm}$.**