The maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is

### Concept & Logic The problem asks for the largest group size that can equally divide two different quantities without leaving a remainder. This perfectly defines the Highest Common Factor (H.C.F.) of the two quantities. $$ \text{Max Students} = \text{H.C.F.}(\text{Number of Pens}, \text{Number of Pencils}) $$ ### Step-by-Step Solution * **Given:** Number of pens = 1001, Number of pencils = 910. * **Calculation / Deduction:** We need to find H.C.F.(1001, 910). * Find the difference between the two values: $1001 - 910 = 91$. * The H.C.F. must be 91 or a factor of 91. Let's check if 91 directly divides both numbers. * $910 \div 91 = 10$ (Perfect division). * Since $1001 = 910 + 91$, it follows that $1001 \div 91 = 11$ (Perfect division). * Since both divide perfectly, the H.C.F. is 91. ### Exam Strategy & Shortcut The difference method is the fastest way to find H.C.F. When you see 1001 and 910, mentally subtract them to get 91. Since 91 is one of the options (a), and it clearly divides 910 ($91 \times 10$), it is the immediate answer. No long division or complex factorization is needed. ### Common Pitfall A frequent mistake is adding the two quantities together ($1001 + 910 = 1911$) and selecting option (d), mistakenly thinking "distribution" implies pooling all resources together. The items must be distributed from their separate pools. ### Final Answer Therefore, the correct answer is **91**.