Three persons go for a morning walk together and start stepping off at the same time. Their steps measure 75 cm, 80 cm, and 85 cm respectively. What is the minimum distance each should walk so that they all cover the distance in a whole number of steps?

Introduction / Context:
This question tests your understanding of the least common multiple (LCM) in a practical context involving step lengths. Each person has a different step length, and we are asked to find the smallest distance that all three can cover in an exact number of steps. Such questions link number theory concepts like multiples and the LCM to real life scenarios, which is common in competitive exams.

Given Data / Assumptions:
- Step length of person 1 = 75 cm.
- Step length of person 2 = 80 cm.
- Step length of person 3 = 85 cm.
- All start walking together and we want a minimum distance that is an exact multiple of each step length.
- Distance is to be reported in metres and centimetres or just metres where appropriate.

Concept / Approach:
The distance each person covers is equal to number of steps multiplied by his or her step length. For each person to complete the same distance in a whole number of steps, the total distance must be a common multiple of all three step lengths. The smallest such distance is their least common multiple (LCM). After computing the LCM in centimetres, we convert that distance into metres (and centimetres if needed) and then match it with the given options.

Step-by-Step Solution:
Step 1: Write down the step lengths: 75 cm, 80 cm, and 85 cm.Step 2: Factorise the numbers if needed.75 = 3 * 5^2, 80 = 2^4 * 5, and 85 = 5 * 17.Step 3: The LCM is found by taking each prime factor at the highest power that appears.LCM = 2^4 * 3 * 5^2 * 17.Compute step by step: 2^4 = 16, 5^2 = 25.So LCM = 16 * 3 * 25 * 17.16 * 3 = 48, 25 * 17 = 425.Now 48 * 425 = (50 - 2) * 425 = 50 * 425 - 2 * 425 = 21250 - 850 = 20400 cm.Step 4: Convert 20400 cm to metres.Since 100 cm = 1 m, 20400 cm = 20400 / 100 = 204 m.

Verification / Alternative check:
Check if 204 m is divisible by each step length. Convert 204 m to centimetres: 20400 cm. Now 20400 / 75 = 272 steps, 20400 / 80 = 255 steps, and 20400 / 85 = 240 steps. All these quotients are whole numbers, so each person will indeed take an integer number of steps to cover 204 m. This confirms that 204 m is the smallest such distance because it is the LCM of the three step lengths.

Why Other Options Are Wrong:
- 222 m 44 cm, 201 m 21 cm, and 208 m do not correspond to the LCM of 75, 80, and 85 when converted to centimetres and checked for divisibility.
- They either leave remainders when divided by one or more of the step lengths or are multiples of a larger common multiple, not the least common multiple.

Common Pitfalls:
Many students confuse greatest common divisor (GCD) with least common multiple (LCM). For this type of problem, always remember that when you want the smallest common size that can accommodate all step lengths without remainder, you should use the LCM. Errors in prime factorisation and arithmetic multiplications can also lead to wrong answers, so it is helpful to redo the multiplication step carefully or break it into smaller parts as shown.

Final Answer:
The minimum distance each should walk is 204 m.