A palace has 18,000 gold coins, 9,600 silver coins, and 3,600 bronze coins. All rooms must contain the same number of coins, and each room can only contain coins of a single type. What is the minimum number of rooms required?

Introduction:
This problem applies the concept of highest common factor (HCF) in a practical distribution scenario. The palace has three types of coins, and each room should hold an equal number of coins, but only one type per room. We must minimise the number of rooms while respecting these rules.

Given Data / Assumptions:

• Gold coins = 18,000
• Silver coins = 9,600
• Bronze coins = 3,600
• Each room contains coins of only one type.
• Every room must contain the same number of coins, say K.
• We need the minimum number of rooms.

Concept / Approach:
If each room contains K coins and K is the same for all rooms, then K must divide 18,000, 9,600, and 3,600 exactly. To minimise the number of rooms, we should maximise K, which means K must be the HCF of the three counts. Once K is known, we can find the number of rooms by dividing each total by K and adding them up.

Step-by-Step Solution:
Find HCF of 18,000, 9,600, and 3,600. Factor 18,000 = 180 × 100 = 18 × 10 × 100 = 2^4 × 3^2 × 5^3 Factor 9,600 = 96 × 100 = 2^7 × 3 × 5^2 Factor 3,600 = 36 × 100 = 2^4 × 3^2 × 5^2 HCF is product of common primes with smallest exponents: For 2: min power = 2^4 For 3: min power = 3^1 For 5: min power = 5^2 So HCF = 2^4 × 3 × 5^2 = 16 × 3 × 25 = 1200 Thus each room should contain K = 1200 coins. Number of rooms for gold = 18,000 ÷ 1200 = 15 Number of rooms for silver = 9,600 ÷ 1200 = 8 Number of rooms for bronze = 3,600 ÷ 1200 = 3 Total rooms = 15 + 8 + 3 = 26

Verification / Alternative check:
We can check that no larger value than 1200 can divide all three numbers. Increasing K would make at least one division non integer. Since 1200 is the HCF, it is the largest possible K, which ensures the minimum possible number of rooms. The computed 26 rooms therefore satisfies all conditions.

Why Other Options Are Wrong:
Choosing a smaller K, such as 600, would increase the number of rooms. That would contradict the requirement of minimum rooms. The options 12, 18, and 24 correspond to such smaller K values and therefore are not optimal. Only 26 matches the HCF based calculation.

Common Pitfalls:
A common error is to find the least common multiple (LCM) instead of the HCF, which has no meaning here. Another mistake is to divide by an arbitrary common factor instead of the highest one, leading to a higher number of rooms than necessary.

Final Answer:
The minimum number of rooms required is 26.