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?

$7$

$10$

$11$

$12$

$19$

Solution:

At row $i,$ column $j,$ Horace writes $91(i - 1) + j$ and Vera writes $141(j - 1) + i.$ Set them equal and simplify to get $9i - 14j = -5,$ so $i = \tfrac{14j - 5}{9},$ an integer exactly when $j \equiv 1 \pmod 9.$ For $j = 1, 10, 19, \ldots, 91,$ that's $11$ values, and $i$ runs $1, 15, 29, \ldots, 141,$ all within range. So $11$ squares match. Thus, C is the correct answer.

2025 AMC 10B Problem 23

Solution:

Problem 23 in Other Years