A solid cuboid has dimensions 65 centimetres by 26 centimetres by 3.9 centimetres. Identical solid cubes of the largest possible size are cut from this cuboid. What is the total surface area, in square centimetres (cm^2), of all the small cubes taken together?

Introduction / Context:
This question combines number theory (greatest common divisor) with three dimensional geometry. A rectangular cuboid is cut into identical cubes of the largest possible size, and we are asked to find the total surface area of all those cubes together. The largest possible cube edge length must exactly divide each of the three dimensions of the cuboid. Once this edge is known, we can determine the number of cubes and then their combined surface area.

Given Data / Assumptions:

• Dimensions of the cuboid: length = 65 cm, breadth = 26 cm, height = 3.9 cm.
• Identical cubes of the largest possible size are cut from the cuboid, with no leftover material.
• We must find the total surface area of all small cubes together.
• The edge length of each small cube must exactly divide all three dimensions.

Concept / Approach:
The largest possible cube that can fit exactly inside the cuboid will have edge length equal to the greatest common divisor (GCD) of the three dimensions, expressed in consistent units. It is convenient to convert all dimensions to millimetres to avoid decimals and find the GCD. After finding the cube edge length, we compute the number of cubes as (volume of cuboid) / (volume of one cube). The surface area of one cube is 6 * a^2, so the total surface area is number of cubes times 6 * a^2.

Step-by-Step Solution:
Step 1: Convert dimensions to millimetres to avoid decimals: 65 cm = 650 mm, 26 cm = 260 mm, 3.9 cm = 39 mm. Step 2: Find the GCD of 650, 260 and 39. Factorising: 650 = 2 * 5^2 * 13, 260 = 2^2 * 5 * 13, 39 = 3 * 13. The common factor is 13. Step 3: So the largest cube edge length is 13 mm, which is 1.3 cm. Step 4: Volume of the cuboid in cubic millimetres is 650 * 260 * 39 = 6,591,000 mm^3. Step 5: Volume of one cube with edge 13 mm is 13^3 = 2197 mm^3. Step 6: Number of cubes n = 6,591,000 / 2197 = 3000. Step 7: Convert cube edge back to centimetres: a = 1.3 cm. Surface area of one cube is 6 * a^2 = 6 * 1.3^2 = 6 * 1.69 = 10.14 cm^2. Step 8: Total surface area of all cubes = n * 6 * a^2 = 3000 * 10.14 = 30,420 cm^2.

Verification / Alternative check:
We can compute the cuboid volume in cubic centimetres as 65 * 26 * 3.9 = 6591 cm^3. The volume of one cube in cubic centimetres is 1.3^3 = 2.197 cm^3. Then the number of cubes is 6591 / 2.197 = 3000, confirming our earlier count. Using 3000 cubes and surface area 10.14 cm^2 per cube again yields 30,420 cm^2. Both millimetre based and centimetre based calculations match, verifying the correctness of the answer.

Why Other Options Are Wrong:
Values such as 15210, 20280, 16440 and 22860 arise from mistaken GCDs (for example using 1.3 incorrectly), miscounting the number of cubes, or omitting the factor 6 when calculating the surface area of one cube. Only 30420 matches the product of 3000 and 10.14 and is consistent with the exact division of the cuboid into cubes of edge 1.3 cm.

Common Pitfalls:
A common error is to treat the cube edge as an integer in centimetres, such as 1 cm or 2 cm, without converting the decimal dimension 3.9 properly. Another pitfall is to forget to convert units to a common scale before taking the GCD or to simply approximate the GCD visually. Finally, some learners compute the total surface area as n * a^2 rather than n * 6 * a^2. Taking care with unit conversion, factorisation and the surface area formula avoids these mistakes.

Final Answer:
The total surface area of all the small cubes together is 30420 cm^2.