If the L.C.M. of three numbers is 9570, then their H.C.F. can be

### Concept & Formula The fundamental rule relating Highest Common Factor (H.C.F.) and Lowest Common Multiple (L.C.M.) of any set of numbers is that the H.C.F. must always be a perfect divisor of the L.C.M. $$ \text{L.C.M.} = k \times \text{H.C.F.} $$ where $k$ is a positive integer. ### Step-by-Step Solution * **Given:** L.C.M. = 9570. * **Calculation / Deduction:** We must test which of the given options perfectly divides 9570 without leaving a remainder. * Testing 11: $9570 / 11 = 870$ (Exact integer). * Testing 12: $9570 / 12 = 797.5$ (Decimal). * Testing 19: $9570 / 19 \approx 503.68$ (Decimal). * Testing 21: $9570 / 21 \approx 455.71$ (Decimal). ### Exam Strategy & Shortcut Use divisibility rules to immediately spot the answer without full long division. For 11, check the alternating sum of digits: $(9 + 7) - (5 + 0) = 16 - 5 = 11$. Since 11 is perfectly divisible by 11, the number 9570 is as well. You can confidently select option (a) and move on in seconds. ### Common Pitfall Students often think they need to find the specific three numbers first to determine the H.C.F. This is impossible with the given data and leads to a dead end. Always rely on the primary relationship rule between L.C.M. and H.C.F. ### Final Answer Therefore, the correct answer is **11**.