Answer :
To solve this problem, let's begin by understanding the distribution of the numbers 1 through 81 and how factors of 3 are involved.
### Step 1: Identify Multiples of 3
From the set of numbers {1, 2, 3, ..., 81}, any multiple of 3 will contribute to a product being divisible by 3. Let's count these multiples.
Numbers divisible by 3 in the range 1 to 81:
[tex]\[ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81 \][/tex]
There are 27 such numbers since:
[tex]\[ 3 \times k \quad \text{(for } k = 1 \text{ to } 27) \][/tex]
### Step 2: Calculate the Placement in Grid
In a 9×9 grid, there are 9 rows and 9 columns.
To ensure the least number of rows and columns contain multiples of 3, we need to distribute the 27 multiples of 3 as efficiently as possible.
### Step 3: Strategy for Minimizing Rows and Columns
We aim to handle this scenario by considering the possible intersections:
1. Minimize Rows:
- If we place 3 multiples of 3 in a row, we can cover 9 rows using exactly [tex]\(\frac{27}{3} = 9\)[/tex] rows.
2. Minimize Columns with Minimal Overlap:
- Then, for columns, we want to place these multiples in such a way as to intersect with minimum additional coverage beyond rows. If each column intersects with, say, 3 such rows with no overlap, we would still need additional columns.
### Step 4: Balancing Rows and Columns
A better optimized arrangement would be to balance the distribution:
- Optimal Numbering of Rows and Columns Needed:
- We need each of 9 rows to contain at least one multiple of 3 to meet our requirement - ensuring the least rows.
- We also need some columns to cover those multiples effectively.
### Step 5: Calculate Possible Minimum
Balancing correctly:
1. If we ensure 3 multiples per every 3 horizontally consecutive rows (thus 3 filled rows), spreading to minimize added columns:
- E.g., placing multiples in a triangular pattern in groups will require only about 3 minimum distinct columns additionally to avoid overlap yet ensure coverage.
### Conclusion
Each of the above calculations show the complexity reductions lead ultimately to essentially:
- The least at minimum 6 rows (with additional optimal column reduction trying balancing but typically intersect minimum will involve typically 6 also).
Thus:
The least number of rows and columns that could have a product divisible by 3 is 6 rows and 6 columns.
This result maintains the balance and achieves minimum feasible intersection needs, ensuring at least each containing multiple of 3 for grid optimality.
### Step 1: Identify Multiples of 3
From the set of numbers {1, 2, 3, ..., 81}, any multiple of 3 will contribute to a product being divisible by 3. Let's count these multiples.
Numbers divisible by 3 in the range 1 to 81:
[tex]\[ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81 \][/tex]
There are 27 such numbers since:
[tex]\[ 3 \times k \quad \text{(for } k = 1 \text{ to } 27) \][/tex]
### Step 2: Calculate the Placement in Grid
In a 9×9 grid, there are 9 rows and 9 columns.
To ensure the least number of rows and columns contain multiples of 3, we need to distribute the 27 multiples of 3 as efficiently as possible.
### Step 3: Strategy for Minimizing Rows and Columns
We aim to handle this scenario by considering the possible intersections:
1. Minimize Rows:
- If we place 3 multiples of 3 in a row, we can cover 9 rows using exactly [tex]\(\frac{27}{3} = 9\)[/tex] rows.
2. Minimize Columns with Minimal Overlap:
- Then, for columns, we want to place these multiples in such a way as to intersect with minimum additional coverage beyond rows. If each column intersects with, say, 3 such rows with no overlap, we would still need additional columns.
### Step 4: Balancing Rows and Columns
A better optimized arrangement would be to balance the distribution:
- Optimal Numbering of Rows and Columns Needed:
- We need each of 9 rows to contain at least one multiple of 3 to meet our requirement - ensuring the least rows.
- We also need some columns to cover those multiples effectively.
### Step 5: Calculate Possible Minimum
Balancing correctly:
1. If we ensure 3 multiples per every 3 horizontally consecutive rows (thus 3 filled rows), spreading to minimize added columns:
- E.g., placing multiples in a triangular pattern in groups will require only about 3 minimum distinct columns additionally to avoid overlap yet ensure coverage.
### Conclusion
Each of the above calculations show the complexity reductions lead ultimately to essentially:
- The least at minimum 6 rows (with additional optimal column reduction trying balancing but typically intersect minimum will involve typically 6 also).
Thus:
The least number of rows and columns that could have a product divisible by 3 is 6 rows and 6 columns.
This result maintains the balance and achieves minimum feasible intersection needs, ensuring at least each containing multiple of 3 for grid optimality.
