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There are three rectangles which contain $100$ squares: $1 \times 100$, $4 \times 11$ and $5 \times 8$
Width: Height:

1

2

3

4

5

6

7

1  1  2  3  4  5  6  7 
2  2  5  8  11  14  17  20 
3  3  8  14  20  26  32  38 
4  4  11  20  30  40  50  60 
5  5  14  26  40  55  70  85 
6  6  17  32  50  70  91  112 
7  7  20  38  60  85  112  140 
If the height is $1$, increasing the width by $1$ increases the number of squares by $1$. This is because the only size of square that we can make is $1 \times 1$, so adding $1$ to the width adds just $1$ square. Eventually, when the rectangle is $1 \times 100$ there will be $100$ rectangles.
If the height is $2$, increasing the width by $1$ increases the number of squares by $3$. This is because when we add $1$ to the width, we can make $2$ additional $1 \times 1$ squares and $1$ additional $2 \times 2$ square. This is a total of $3$ extra squares.
If we continue the pattern $2, 5, 8, \ldots$ in the height of $2$ row (all $1$ less than multiples of $3$), we eventually get to $\ldots, 98, 101, \ldots$  missing out $100$. This tells us that it isn't possible to make a rectangle with $100$ squares when the height (or width) is $2$.
If the height is $3$, increasing the width by $1$ allows us to make $3$ more $1 \times 1$ squares, $2$ more $2 \times 2$ squares and $1$ more $3 \times 3$ square. This is a total of $6 = 3 + 2 + 1$ squares. This gives us the sequence $14, 20, 26, 32, \ldots, 98, 104, \ldots$ (all $2$ more than multiples of $6$) which tells us that we can't make a rectangle with exactly $100$ squares when the
height (or width) is $3$.
Using the same reasoning for height of $4$, we see that increasing the width by $1$ increases the number of squares by $10$, and that we can make rectangles with $20, 30, \ldots, 90, 100, 110, \ldots$ squares. $100$ is in this list! In fact, a $4 \times 11$ rectangle contains exactly $100$ squares.
We can repeat this for the other heights (or widths) in the table to find all the rectangles with exactly $100$ squares. These are $1 \times 100$, $4 \times 11$ and $5 \times 8$.