How to find the minimal DFA for the language? Design a minimal DFA for the language which contains numbers in the form of binary strings starting with $101_2$ and is divisible by $100_{10}$?

I think the numbers will be 700,1400,2100 ... in the form of binary strings which are starting with $101_2$ and are even divisible by $100_{10}$.

100 is divisible by 25 and 4.

For divisible by 4 - Minimum 3 state DFA is possible For divisible by 25 - Min 25 state DFA is possible.

Hence, for divisible by $100_{10}$ - Minimum 100 states DFA is possible .

So, for starting with $101_2$, I have to add 3 extra states along with one dead state.

So, total states in minimal DFA = 100 + 3 + 1 = 104

Is my approach correct or am I missing something?
Your approach towards estimating the number of states is just fine. however the number of states looks quite intimidating and it may be cumbersome to minimize the DFA later.
 In such cases constructing the automaton through intersection is easier. for example as we know the regular languages are closed under the intersection operation we can devise 3 different automata to accept :
 1. String divisible by 25
 2. String divisible by 4
 3. Strings beginning with 101

And then arrive at the final automaton by taking intersections of the 3 machines.

How this helps?

Many of the states originally present in the different machines gradually become unreachable from the initial state in the final automaton and thus the number of states reduce or the DFA is minimized.