Can you solve it without using CRT?

I know,I know this problem can be easily killed using Chinese Remainder Theorem but I don't know this theorem and hence solved it in this way: First of all,let us say that, $2^{999}=100*x+a$ .Now,let us divide the equation by $2^2$ ,it becomes, $2^{997}=25x+\dfrac{a}{4}$ .Now,if we find the remainder when $2^{997}$ is divided by $25$ we would get $\dfrac{a}{4}$ and thus, $a$ .For finding that consider this: $2^5\equiv 7\pmod{25}\\ \Rightarrow 2^{10}\equiv 49\equiv(-1)\pmod{25}\\ \Rightarrow 2^{990}\equiv (-1)\equiv 24\pmod{25}\\ \Rightarrow 2^{997}\equiv (24*128)\equiv 3072\equiv 22\pmod{25}$ .That implies $\dfrac{a}{4}=22\Rightarrow a=22*4=88$ .