Math Gold Medalist

Lor

2024 AMC 8 

Problem 16

Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by $3$?

$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$

Considering Range

Maximizing Product When Sum is Constant

By finding a good example prove that answer<=11

Assume that multiples of 3 are in a columns and b rows then answers<=ab

Use contradiction to prove answer>=11

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   Solution