Three tanks have capacities of 98 litres, 182 litres, and 266 litres respectively. Find the capacity of the largest measuring cylinder that can be used to measure out and exactly fill each tank an integer number of times, without leaving any remainder in any tank.

Introduction / Context:
This question is a real life application of the highest common factor (HCF). When we want one measuring container to exactly measure several different quantities, the largest possible size of that container is the HCF of the given capacities. It ensures that each capacity is a whole number multiple of the measuring cylinder without any leftover liquid.

Given Data / Assumptions:

• Tank 1 capacity = 98 litres
• Tank 2 capacity = 182 litres
• Tank 3 capacity = 266 litres
• We want the largest possible measuring cylinder that measures each capacity exactly.

Concept / Approach:
The highest common factor of the three capacities is the largest integer that divides each of them exactly. We find the HCF by prime factorization of each capacity and then taking the product of all common primes with their smallest exponent across the three numbers.

Step-by-Step Solution:
98 = 2 * 7^2 182 = 2 * 7 * 13 266 = 2 * 7 * 19 Common prime factors in all three: 2^1 and 7^1. HCF = 2 * 7 = 14 litres. Thus, a 14 litre cylinder will exactly measure each tank: 98 / 14 = 7 times, 182 / 14 = 13 times, and 266 / 14 = 19 times.

Verification / Alternative check:
Checking divisibility: 98 mod 14 = 0, so 14 divides 98 exactly. 182 mod 14 = 0, so 14 divides 182 exactly. 266 mod 14 = 0, so 14 divides 266 exactly. No larger option among the choices divides all three capacities exactly, so 14 litres is indeed the largest possible measure.

Why Other Options Are Wrong:
7 litres divides all capacities, but it is not the largest possible measure. 21, 42, and 98 litres do not divide each of 98, 182, and 266 without leaving a remainder, so they cannot be used as the required measuring cylinder capacity.

Common Pitfalls:
Common errors include choosing a common divisor that is not the greatest one, or confusing HCF with LCM. Here we must maximize the cylinder size, which is why we focus on the HCF and not on the least common multiple of the capacities.

Final Answer:
14 litres