The product of the L.C.M. and H.C.F. of two numbers is 24. The difference of two numbers is 2. Find the numbers.

### Concept & Logic The foundational rule of H.C.F. and L.C.M. states that the product of the Highest Common Factor and the Lowest Common Multiple is exactly equal to the product of the two numbers themselves. $$ \text{Product of Numbers} = \text{H.C.F.} \times \text{L.C.M.} $$ ### Step-by-Step Solution * **Given:** * $\text{H.C.F.} \times \text{L.C.M.} = 24$ * Difference of the two numbers = 2 * Let the two numbers be $x$ and $y$ (where $x > y$). * From the foundational rule, $x \times y = 24$. * We are given that $x - y = 2$, which means $x = y + 2$. * Substitute $x$ into the product equation: $(y + 2) \times y = 24$. * Expand to form a quadratic equation: $y^2 + 2y - 24 = 0$. * Factor the quadratic equation: $(y + 6)(y - 4) = 0$. * Since numbers in this context are positive integers, $y = 4$. * If $y = 4$, then $x = 4 + 2 = 6$. * The numbers are 6 and 4. ### Exam Strategy & Shortcut **Option Elimination** is by far the fastest method here. Since $\text{Product of Numbers} = \text{H.C.F.} \times \text{L.C.M.}$, we know the product of the two numbers must be 24. Let's test the options: * (a) $2 \times 4 = 8$ (Incorrect) * (b) $6 \times 4 = 24$ (Correct, and $6 - 4 = 2$) * (c) $8 \times 6 = 48$ (Incorrect) * (d) $8 \times 10 = 80$ (Incorrect) You can arrive at option (b) in under 5 seconds without writing a single algebraic equation. ### Common Pitfall Wasting time solving the quadratic equation algebraically is a classic pitfall. While mathematically sound, standard algebraic methods are too slow for competitive exams when a simple cross-check of the options takes mere seconds. Always look at the options first when given simple products and differences. ### Final Answer **Therefore, the correct answer is 6 and 4.**