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 with the numbers 1 through 141×91=12,831. Horace fills horizontally (left to right within each row); Vera fills vertically (top to bottom within each column). 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