The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is

### Concept & Logic When two numbers have an H.C.F. of $H$, the numbers can be represented as $Hx$ and $Hy$, where $x$ and $y$ are co-prime factors. The L.C.M. of these numbers is given by: $$ \text{L.C.M.} = H \times x \times y $$ The problem explicitly provides $H$ and the other two factors ($x$ and $y$). ### Step-by-Step Solution * **Given:** H.C.F. = 23, and the other two factors of the L.C.M. are 13 and 14. * **Calculation / Deduction:** Let the two numbers be $N_1$ and $N_2$. * Based on the concept, the numbers are $23 \times 13$ and $23 \times 14$. * The problem asks for the larger of the two numbers. * Larger number = $23 \times 14$. * $23 \times 14 = 322$. ### Exam Strategy & Shortcut You don't need to calculate both numbers. Simply identify the larger factor (14) and multiply it by the H.C.F. (23). To quickly calculate $23 \times 14$, use the unit digit trick: $3 \times 4 = 12$, so the answer must end in 2. Looking at the options, only (c) 322 ends in 2, making it an instant pick. ### Common Pitfall A common mistake is multiplying all three numbers together ($23 \times 13 \times 14$) assuming the question asks for the L.C.M., or mistakenly assuming 13 and 14 are the numbers themselves rather than the co-prime factors. Read the prompt carefully. ### Final Answer Therefore, the correct answer is **322**.