Question

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 $141 \times 91 = 12{,}831$ into the squares. Horace fills the grid horizontally: he puts $1$ through $91$ in order from left to right into row $1,$ puts $92$ through $182$ into row $2$ in order from left to right, and continues similarly through row $141.$ Vera fills the grid vertically: she puts $1$ through $141$ in order from top to bottom into column $1,$ then $142$ through $282$ into column $2$ in order from top to bottom, and continues similarly through column $91.$ How many squares get two copies of the same number?

$7$

$10$

$11$

$12$

$19$

Answer: C

Concepts:Diophantine Equation modular arithmetic counting integers in a range

Difficulty rating: 2040

Small Hint:

Write the Horace-number and the Vera-number of the square in row $r,$ column $c$ as formulas in $r$ and $c$

Big Hint:

Setting them equal gives $9r = 14c - 5;$ find which $c$ make $r$ an integer in range

Solution:

In row $r,$ column $c,$ Horace writes $(r-1)\cdot 91 + c$ and Vera writes $(c-1)\cdot 141 + r.$ Setting these equal gives $90r - 140c = -50,$ i.e. $9r = 14c - 5.$ This requires $c \equiv 1 \pmod 9,$ so $c = 1, 10, 19, \ldots, 91$ — that is $11$ values, and each yields a valid $r$ between $1$ and $141.$ So $11$ squares match.

Thus, the correct answer is C.

2025 AMC 12B Problem 19

Answer: C

Difficulty rating: 2040

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years