Find the maximum possible number of students among whom 1001 pens and 910 pencils can be distributed such that: • Each student gets the same number of pens, and • Each student gets the same number of pencils. What is the greatest possible number of students?

Introduction / Context:
This problem is about equal distribution of two different items. If the same number of pens and the same number of pencils must be given to each student with no remainder, then the number of students must divide both totals exactly. The largest such number is the HCF (GCD) of the two quantities.

Given Data / Assumptions:

• Total pens = 1001
• Total pencils = 910
• Each student must receive the same integer number of pens
• Each student must receive the same integer number of pencils

Concept / Approach:
If the number of students is S, then S must divide 1001 and 910. Therefore:
S = gcd(1001, 910)
This gcd gives the maximum possible number of students for which a perfect distribution is possible.

Step-by-Step Solution:
Step 1: Compute gcd(1001, 910).1001 - 910 = 91, so 1001 mod 910 = 91.910 mod 91 = 0, because 91 * 10 = 910.Therefore, gcd(1001, 910) = 91.Step 2: So the maximum possible number of students is S = 91.

Verification / Alternative check:
For 91 students: pens per student = 1001 / 91 = 11 and pencils per student = 910 / 91 = 10. Both are integers, so the distribution works perfectly. Any number greater than 91 would fail to divide at least one of the quantities.

Why Other Options Are Wrong:
101: 910 is not divisible by 101.130: 1001 is not divisible by 130.910: 1001 is not divisible by 910.1001: 910 is not divisible by 1001.

Common Pitfalls:
Choosing the smaller of the two numbers or their difference without computing the gcd.Confusing HCF with LCM in equal distribution problems.Not checking divisibility for both items (pens and pencils) simultaneously.

Final Answer:
91