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last digit of number in cell (2021, 2021) 2021 Malaysia IMONST 1 Senior P20

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September 22, 2021
number theoryDigitsLast digit

Problem Statement

The cells of a 2021×20212021\times 2021 table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate (1,1)(1, 1), contains the number 00. For every other cell CC, we consider a route from (1,1)(1, 1) to CC, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell CC. For example, the cell with coordinate (2,1)(2, 1) contains 1=1+01 = 1 + 0, the cell with coordinate (3,1)(3, 1) contains 3=2+0+13 = 2 + 0 + 1, and the cell with coordinate (3,2)(3, 2) contains 7=3+0+1+37 = 3 + 0 + 1 + 3. What is the last digit of the number in the cell (2021,2021)(2021, 2021)?
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