The H.C.F. and L.C.M. of two numbers are 50 and 250 respectively. If the first number is divided by 2, the quotient is 50. The second number is

### Concept & Logic The relationship between any two numbers and their Highest Common Factor (H.C.F.) and Lowest Common Multiple (L.C.M.) is a constant product. Additionally, we use the basic division rule to find the unknown first number. $$ \text{First Number} \times \text{Second Number} = \text{H.C.F.} \times \text{L.C.M.} $$ $$ \text{Dividend} = \text{Divisor} \times \text{Quotient} $$ ### Step-by-Step Solution * **Given:** * H.C.F. = 50 * L.C.M. = 250 * First, determine the value of the first number. The problem states that when it is divided by 2, the quotient is 50. * $\text{First Number} = 2 \times 50 = 100$. * Now, substitute the known values into the fundamental product formula: * $100 \times \text{Second Number} = 50 \times 250$ * Simplify the right side of the equation: * $100 \times \text{Second Number} = 12500$ * Isolate the second number: * $\text{Second Number} = \frac{12500}{100} = 125$. ### Exam Strategy & Shortcut Avoid calculating large numbers like 12500. Set up the equation: $100 \times N = 50 \times 250$. Notice the relationship between the numbers on both sides. $100$ is exactly twice $50$. For the equation to balance out, $N$ must be exactly half of $250$. Half of 250 is 125. You can solve this entirely in your head in under 10 seconds. ### Common Pitfall Students frequently misread the phrase "divided by 2, the quotient is 50" and hastily assume the first number itself is 50. This trap leads them to calculate $\frac{50 \times 250}{50} = 250$, incorrectly choosing option (d). Always calculate the initial dividend carefully. ### Final Answer **Therefore, the correct answer is 125.**