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Problem 212

Algebra Difficulty 1.8 Find the answer CEMC Pascal

Morgan uses a spreadsheet to create a table of values. In the first column, she lists the positive integers from 1 to 400. She then puts integers in the second column in the following way: if the integer in the first column of a given row is $n$ , the number in the second column of that row is $3 n+1$ . Which of the following integers does not appear in the second column: 31, 94, 131, 331, 907?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since $31=3 imes 10+1$ and $94=3 imes 31+1$ and $331=3 imes 110+1$ and $907=3 imes 302+1$ , then each of $31,94,331$ , and 907 appear in the second column of Morgan's spreadsheet. Thus, 131 must be the integer that does not appear in Morgan's spreadsheet. (We note that 131 is 2 more than $3 imes 43=129$ so is not 1 more than a multiple of 3.)

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