Comparing Products: Insights into the Smallest and Largest Products

In this article, we will explore the comparison of three specific products:

997 * 998 * 999 998 * 998 * 998 996 * 999 * 999

By performing detailed calculations and logical reasoning, we will determine which of these products is the smallest and which is the largest.

Calculations and Reasoning

Calculation of 997 * 998 * 999

First, let's compute the product 997 * 998 * 999.

Calculate 998 * 999:

998 * 999  998 * (1000 - 1)  998000 - 998  997002

Multiply 997 by the result:

997 * 997002  997^2 * 999 ≈ 994009 * 999 ≈ 996005994

Calculation of 998 * 998 * 998

Next, let's compute the product 998 * 998 * 998.

Cube of 998:

998^3  1000 - 2^3  1000^3 - 3 * 1000^2 * 2   3 * 1000 * 2^2 - 2^3
       1000000000 - 6000000   12000 - 8
       994012000

Calculation of 996 * 999 * 999

Lastly, let's compute the product 996 * 999 * 999.

Calculate 999^2:

999^2  998001

Multiply 999^2 by 996:

996 * 998001  996 * (1000^2 - 1)  996000000 - 996000 - 996
           ≈ 994008996

Conclusion

Now we compare the three calculated values:

997 * 998 * 999 ≈ 996005994 998 * 998 * 998 ≈ 994012000 996 * 999 * 999 ≈ 994008996

The smallest number is 996 * 999 * 999 ≈ 994008996 and the largest number is 997 * 998 * 999 ≈ 996005994.

Intuitive Explanation

Intuitive Answer: Square vs. Rectangle

To gain an intuitive understanding, let's explore the concept through a geometric analogy. Imagine a square and a rectangle with the same perimeter. If you make one corner of the rectangle match the square, you can consider the two zones that are in one figure but not the other.

The zone in the square but not in the rectangle has a length equal to the square's side. The zone in the rectangle but not in the square has a length equal to the rectangle's width, which is smaller. Therefore, the zone in the square is always smaller, indicating the square has a larger area.

This intuition can be extended to three dimensions. If we have a cuboid with a fixed surface area, the cube has the maximum volume. In our specific problem, the sum of the factors (997 998 999) is the same for all products. Thus, the product with all factors equal (998 * 998 * 998) will be the largest, and the product with the smallest differences between factors will be the smallest.