1998 AJHSME Problems/Problem 24
A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, 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?
The numbers that are shaded are the triangular numbers, which are numbers in the form for positive integers. They can also be generated by starting with , and adding as in the description of the problem.
Squares that have the same remainder after being divided by will be in the same column. Thus, we want to find when the last remainder, from to , is found.
So, instead of adding , we can effectively either add or subtract if we are only concerned about remainders when divided by . We will pick the number that keeps the terms on the list between and . We get:
Finally, a term with is found, and checking, all numbers through are also on the right side of the list. This means the last term in our sequence is the first time that column is shaded. There are terms in the sequence, leading to an answer of , which is choice .
Note that the triangular numbers up to are . When you divide each of those numbers by , all remainders must be present. We first search for number(s) that are evenly divisible by ; if two such numbers exist, we search for numbers that leave a remainder of , etc.
Quickly scanning the list, only and are even. That smaller list doesn't have any multiples of until it hits . So must be the answer.
The numbers shaded are triangular numbers of the form . For this number to be divisible by , the numerator must be divisible by . Since only one of and can be even, only one of them can have factors of . Therefore, the first time the whole expression is divisible by is when either or when . This gives as the first time is divisible by , which gives . No other triangular number lower than that is divisible by , and thus the column on the checkerboard won't be filled until then. That gives as the right answer.
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