To find the least number of square stones required to pave the rectangular courtyard, we need to determine the largest size of square stones that can completely fit both the length and the breadth of the courtyard without cutting any stone.

This problem can be solved using the greatest common divisor (GCD) of the courtyard's length and breadth.

Given dimensions of the courtyard:

• Length = 2016 cm
• Breadth = 1560 cm

Step 1: Find the GCD of 2016 and 1560

The GCD will give us the side length of the largest square stone that can fit both dimensions.

To find the GCD of 2016 and 1560, we use the Euclidean algorithm:

• Divide 2016 by 1560: $2016 \div 1560 = 1 \, \text{(remainder 456)}$
• Now, divide 1560 by 456: $1560 \div 456 = 3 \, \text{(remainder 192)}$
• Next, divide 456 by 192: $456 \div 192 = 2 \, \text{(remainder 72)}$
• Now, divide 192 by 72: $192 \div 72 = 2 \, \text{(remainder 48)}$
• Finally, divide 72 by 48: $72 \div 48 = 1 \, \text{(remainder 24)}$
• Lastly, divide 48 by 24: $48 \div 24 = 2 \, \text{(remainder 0)}$

So, the GCD of 2016 and 1560 is 24 cm.

Step 2: Calculate the number of stones

Now, we can calculate how many square stones of side length 24 cm are needed.

• Number of stones along the length: $2016 \div 24 = 84$
• Number of stones along the breadth: $1560 \div 24 = 65$

Step 3: Total number of stones

The total number of square stones required is: $84 \times 65 = 5460$

Thus, the least number of square stones required to pave the courtyard is 5460.

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Do you need further details or have any questions?

Here are 5 related questions for further understanding:

1. How would the process change if the courtyard had non-integer dimensions?
2. What if the stones were rectangular instead of square?
3. Can this method work for non-rectangular courtyards?
4. How would using smaller stones affect the number of stones required?
5. What is the advantage of using square stones with side lengths equal to the GCD?

Tip: The Euclidean algorithm is an efficient way to find the GCD, especially when dealing with large numbers.