The numbers 1 to 100 are arranged in a table as shown below. A L-shaped shaded frame can be placed around various numbers throughout the table. $L_n$ is used to refer to the shaded frame that is drawn around a particular number. For example, $L_2$ refers to the shaded frame shown above drawn around the number 2.

1. Identify the largest possible corner position for the same L-shaped frame to fit in the table

The L-shaped frame shown for $L_2$ has its corner at the number $2$ and extends to the right edge and the bottom edge of the $10$ by $10$ table.

If the corner were placed further to the right, the horizontal arm would go beyond the table. Hence the largest possible value of $n$ is

$n=2.$

1. Write the sums of the horizontal and vertical arms of $L_n$

Since the L-frame has the same size as the example, the horizontal arm has 9 consecutive numbers:

$(n)+(n+1)+\cdots+(n+8)$

The vertical arm has 10 numbers increasing by 10:

$n+(n+10)+(n+20)+\cdots+(n+90)$

2. Add the two arm sums, count the corner once only, and simplify

The corner number $n$ is included in both arms, so subtract it once.

$$\begin{aligned} \text{Sum} &= \big[n+(n+1)+\cdots+(n+8)\big] \\ &\quad + \big[n+(n+10)+\cdots+(n+90)\big] - n \\ &= (9n+36)+(10n+450)-n \\ &= 18n+486. \end{aligned}$$

1. Show that the expression for the sum has a factor of 3

From part (b), the sum is
$18n+486$.

Since
$18n+486=3(6n+162)$,
the sum is always divisible by 3.