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    OlympiadMediumNumber Theory5–6

    Every Entry of the Times Table

    Todas as Casas da Tabuada

    A 10×1010 \times 10 table has its rows and its columns labelled 11 to 1010, and each cell holds the product of its row label and its column label. What is the sum of all 100100 entries?

    Add one row at a time and look at what every row has in common before adding the rows together.

    Solution

    Step 1 of 5

    1. 1.Row rr contains r×1,r×2,…,r×10r \times 1, r \times 2, \ldots, r \times 10, so its sum is r×(1+2+⋯+10)=55rr \times (1 + 2 + \cdots + 10) = 55r.