A rectangular grid of squares has 141 rows and 91 columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from 1 through 141×91=12,831 into the squares. Horace fills the grid horizontally: he puts 1 through 91 left to right in row 1, 92 through 182 in row 2, and so on. Vera fills the grid vertically: she puts 1 through 141 top to bottom into column 1, then 142 through 282 into column 2, and so on. How many squares get two copies of the same number?
Answer Choices:
A. 7
B. 10
C. 11
D. 12
E. 19
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Let (x,y) represent the column and row at which the two numbers are equal.
H=91(y−1)+x
V=141(x−1)+y
Equating, we get 91(y−1)+x=141(x−1)+y, or
91y−91+x=141x−141+y⟹140x−90y=50⟹14x−9y=5.
Working mod 9, we get
14x−9y≡5(mod9)⟹5x≡5(mod9)⟹x≡1(mod9).
Let x=9k+1. Then,
14(9k+1)−9y=5⟹9y=14⋅9k+14−5=126k+9⟹y=14k+1.
We require
1≤x≤91⟹1≤9k+1≤91⟹0≤k≤10
1≤y≤141⟹1≤14k+1≤141⟹0≤k≤10.
Thus there are a total of 11 values of k, implying there are (C) 11 cells where both of them write the same number.
The problems on this page are the property of the MAA's American Mathematics Competitions