In the diagram, the grid has 150 rows and 150 columns, numbered from 1 to 150 . 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 $n$th box in row $n$ is shaded. Which column has the greatest number of shaded boxes?
(A) 144
(B) 120
(C) 150
(D) 96
(E) 100

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-073.jpg?height=261&width=293&top_left_y=1298&top_left_x=1363)
In the 1st row, every box is shaded.

In the 2nd row, the boxes whose column numbers are multiples of 2 are shaded.

In the 3rd row, the boxes whose column numbers are multiples of 3 are shaded.

In the $n$th row, the boxes whose column numbers are multiples of $n$ are shaded.

So in a particular column, the boxes which are shaded are those which belong to row numbers which are divisors of the column number.

(We can see this for instance in columns 4 and 6 where the boxes in rows 1,2 and 4, and 1,2,

3 , and 6 , respectively, are shaded.)

So to determine which of the given columns has the largest number of shaded boxes, we must determine which of the given numbers has the greatest number of divisors.

To find the divisors of 144, for instance, it is easier to find the prime factors of 144 first.

To do this, we see that $144=16 \times 9=2 \times 2 \times 2 \times 2 \times 3 \times 3=2^{4} \times 3^{2}$.

We can then write out the divisors of 144 , which are $1,2,3,4,6,8,9,12,16,18,24,36,48$, 72,144 , or a total of 15 divisors.

Similarly, $120=2^{3} \times 3 \times 5$, so we can find the divisors of 120 , which are $1,2,3,4,5,6,8,10$, $12,15,20,24,30,40,60,120$, or a total of 16 divisors.

Since $150=2 \times 3 \times 5^{2}$, the divisors of 150 are $1,2,3,5,6,10,15,25,30,50,75,150$, or a total of 12 divisors.

Since $96=2^{5} \times 3$, the divisors of 96 are $1,2,3,4,6,8,12,16,24,32,48,96$, or a total of 12 divisors.

Since $100=2^{2} \times 5^{2}$, the divisors of 100 are $1,2,4,5,10,20,25,50,100$, or a total of 9 divisors. So the number with the most divisors (and thus the column with the most shaded boxes) is 120 .

ANSWER: (B)