Just do it in 5 seconds!

$\begin{aligned} 9797 \div 97 = \frac {9797}{97} = \frac {97\times\color{#3D99F6}101}{97} & = \color{#3D99F6} 101 \\ 9898 \div 98 = \frac {9898}{98} = \frac {98\times\color{#3D99F6}101}{98} & =\color{#3D99F6} 101 \\ 9999 \div 99 = \frac {9999}{99} = \frac {99\times \color{#3D99F6}101}{99} & =\color{#3D99F6} 101 \\ \\ 100100 \div 100 = \frac {100100}{100} = \frac {100\times\color{#D61F06} 1001}{100} & = \color{#D61F06} \boxed{1001}\end{aligned}$

Let $a$ be any $n$ digit number. Then, $\overline{aa}$ contains $2n$ digits. Let us divide $\overline{aa}$ by $a$ .

If $n = 1$ $\dfrac{\overline{aa}}{a} = \dfrac{10a + a}{a} = \dfrac{10a}{a} + \dfrac{a}{a} = 10 + 1 = \boxed{11}$ If $n = 2$ $\dfrac{\overline{aa}}{a} = \dfrac{100a + a}{a} = \dfrac{100a}{a} + \dfrac{a}{a} = 100 + 1 = \boxed{101}$ If $n = 3$ $\dfrac{\overline{aa}}{a} = \dfrac{1000a + a}{a} = \dfrac{1000a}{a} + \dfrac{a}{a} = 1000 + 1 = \boxed{1001}$ $\large \vdots$

If $n = n$ $\dfrac{\overline{aa}}{a} = \dfrac{10n(a) + a}{a} = \dfrac{10na}{a} + \dfrac{a}{a} = \boxed{\color{#20A900}10n + 1}$ Note : If you are confused with any of these steps see my comment in the comments section where I had explained this more clearly.

So, the value of $\dfrac{\color{#3D99F6}100100}{\color{#D61F06}100} = 10(3) + 1 = \color{#69047E}1001$

$9797=9700+97=97\times100+97=97(100+1)=97\times101$ , and this is same for $98$ and $99$

But, $100100=100000+100=100\times1000+100=100(1000+1)=100\times1001$

I almost chose $101$ at the first second :p

$100100 \div 100 = \frac{100100}{100} = \frac{1001\cancel{0}\cancel{0}}{1\cancel{0}\cancel{0}} = 1001$

The rule is so smooth, but if u can look at the numbers you should realise, that there are always four-digit numbers, but in the question to solve there is the six-digit one. On the first thought, you would probably like to think this is fine - two numbers in your big number are getting bigger, so the whole number should be bigger though. However, the right '1' isn't able to change the whole number - it can only happen if every number on the left from it was '9' because it's written in the decimal system. I hope you can feel what I mean :D