The H.C.F. of two numbers is 8. Which one of the following can never be their L.C.M.?

### Concept & Logic The core rule relating these two metrics is that the Highest Common Factor (H.C.F.) of any given set of numbers must be a perfect factor (divisor) of their Lowest Common Multiple (L.C.M.). If an L.C.M. value is not a multiple of the H.C.F., it is an invalid L.C.M. $$ \text{L.C.M.} = k \times \text{H.C.F.} \text{ (where } k \text{ is an integer)} $$ ### Step-by-Step Solution * **Given:** H.C.F. = 8. * We are looking for an option that is NOT a multiple of 8 (i.e., 8 does not divide it without a remainder). * Let's evaluate the choices: * **Option (a) 24:** $24 / 8 = 3$. (Valid L.C.M. candidate). * **Option (b) 48:** $48 / 8 = 6$. (Valid L.C.M. candidate). * **Option (c) 56:** $56 / 8 = 7$. (Valid L.C.M. candidate). * **Option (d) 60:** $60 / 8 = 7.5$. (Invalid candidate; leaves a remainder of 4). * Because 60 is not perfectly divisible by 8, it can never represent the L.C.M. for this pair of numbers. ### Exam Strategy & Shortcut This question strictly tests your knowledge of multiplication tables. You should know the multiples of 8 by heart: $8, 16, 24, 32, 40, 48, 56, 64$. By rapidly scanning the options against this list, 60 immediately stands out as the odd one out. ### Common Pitfall Similar to other negative-constraint questions, students rushing under time pressure might read "can never be" as "can be" and erroneously select 24 simply because it is the first valid multiple they spot. Always double-check what the question is specifically asking you to identify. ### Final Answer **Therefore, the correct answer is 60.**