Counting the Shaded Squares.

Probability Level 2

In the diagram, the grid has $100$ rows and $100$ columns, numbered from 1 to $100$ . In row $1$ , every box is shaded. In row $2$ , every second box is shaded. In row $3$ , every third box is shaded. The shading continues in this way, so that every $nth$ box in row $n$ is shaded.

Among the given answers, which column has the greatest number of shaded boxes?

50 100 60 30 40

2 solutions

Chew-Seong Cheong
Sep 12, 2018

Let the row number and column number be $r$ and $n$ respectively. We note that $r$ th row has all $n \bmod r = 0$ boxes shaded. This means that the number of boxes shaded in $n$ th column is the number of divisors of $n$ .

Now let us check the numbers of divisors of the answer options.

Since $30 = 2^{\color{#D61F06}1} \times 3^{\color{#D61F06}1} \times 5^{\color{#D61F06}1}$ , then the number of divisors is $({\color{#D61F06}1}+1)({\color{#D61F06}1}+1)({\color{#D61F06}1}+1) = 8$ .
Since $40 = 2^{\color{#D61F06}3} \times 5^{\color{#D61F06}1}$ , then the number of divisors is $({\color{#D61F06}3}+1)({\color{#D61F06}1}+1) = 8$ .
Since $50 = 2^{\color{#D61F06}1} \times 5^{\color{#D61F06}2}$ , then the number of divisors is $({\color{#D61F06}1}+1)({\color{#D61F06}2}+1) = 6$ .
Since $60 = 2^{\color{#D61F06}2} \times 3^{\color{#D61F06}1} \times 5^{\color{#D61F06}1}$ , then the number of divisors is $({\color{#D61F06}2}+1)({\color{#D61F06}1}+1)({\color{#D61F06}1}+1) = 12$ .
Since $100 = 2^{\color{#D61F06}2} \times 5^{\color{#D61F06}2}$ , then the number of divisors is $({\color{#D61F06}2}+1)({\color{#D61F06}2}+1) = 9$ .

Therefore, column $\boxed{60}$ has the greatest number of shaded boxes.

Thank you, nice solution.

Hana Wehbi - 2 years, 9 months ago

Hana Wehbi
Sep 11, 2018

The number that has many divisors is the one with the most shaded squares. This is not a complete solution but a hint on how to solve it.

Counting the Shaded Squares.

2 solutions

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