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, 2023

Number theory Difficulty 2.4 Junior Find the answer Canada

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 $3n+1$ . Which of the
following integers does not appear in the second column?

Pick one

Solution

Since $31 = 3 \times 10 + 1$
and $94 = 3 \times 31 + 1$ and $331 = 3 \times 110 + 1$ and $907 = 3 \times 302 + 1$ , 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 $3 \times 43 = 129$ so is not 1 more than a
multiple of 3.)

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