Problem of the Month (August 2009)

This month we investigate three problems involving products.
1. Consider rectangular grids of digits where the product of the numbers in the rows is equal to the product of the numbers in the columns. Since leading 0's should not exist and trailing 0's are less interesting, we only allow positive digits. We also have excluded grids that contain multiple copies of the same factor. We call these multiplicative grids. For example, the 3×2 rectangular grids are shown below.

135 156 194 567 576
154 295 276 432 444

Can you find some multiplicative grids? What about other bases? What about non-rectangular grids? What if each entry in the grid is k digits?

2. Suppose k, m, and n are positive integers. If we group the integers 1 through mn into m groups of n and multiply the numbers in each group, how close can the sum of k of these products be to the sum of the other m-k products? For example,

7⋅8⋅11 = 1⋅3⋅12 + 2⋅4⋅5 + 6⋅9⋅10
2⋅5⋅6 + 8⋅9⋅11 = 1⋅3⋅4 + 7⋅10⋅12

Unfortunately, sometimes we can only get close. This left-hand side and right-hand side differ by 2:

4⋅5⋅8⋅12 ≈ 1⋅2⋅7⋅10 + 3⋅6⋅9⋅11

What are the best we can do for various values of m, n, and k?

3. For which integers n > 1 and m does the expression P = xn + xn-1 + . . . + x2 + x + m factor over the integers? What infinite families are there? Can you find any other polynomials of this form that factor over the integers without a linear factor, other than the ones below?

x7 + x6 + x5 + x4 + x3 + x2 + x – 2 = (x3 + x – 1)(x4 + x3 + x + 2)
x13 + x12 + x11 + . . . + x3 + x2 + x – 3 = (x3 + x2 – 1)(x10 + x8 + x7 + 2x5 + x3 + 2x2 – x + 3)
x4 + x3 + x2 + x + 12 = (x2 + 3x + 4)(x2 – 2x + 3)
x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 20 = (x4 – x3 + x2 – 3x + 4)(x4 + 2x3 + 2x2 + 4x + 5)
x10 + x9 + x8 + . . . + x3 + x2 + x – 22 = (x2 + x + 2)(x8 – x6 + 2x5 + x4 – 4x3 + 3x2 + 6x – 11)
x5 + x4 + x3 + x2 + x – 55 = (x2 – x + 5)(x3 + 2x3 – 2x – 11)