Problem:
A rectangular board of columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered through , row two is through , and so on. A student shades square , then skips one square and shades square , skip two squares and shades square , ships squares and shades square , and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
Answer Choices:
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Solution:
The numbers in the first column all have remainders of when divided by , those of the second column have remainders of when divided by , and so on. We need to find numbered squares so that each remainder through appears at least once. The squares that are shaded are numbered , , and the remainders upon dividing by are . Thus, we must shaded square to obtain the first shaded square in the last column.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions