Binary Answer To The Ultimate Question

The answer, $x$ , must be divisible by 3, 2, and 7. Since it is divisible by 2, the last digit must be 0. Since it is divisible by 3, the sum of digits (i.e. the number of 1's) must be a multiple of 3. To minimize the answer, let the sum of digits be 3. Next, notice that

$10\equiv 3\quad (mod\quad 7)\\ { 10 }^{ 2 }\equiv 2\quad (mod\quad 7)\\ { 10 }^{ 3 }\equiv 6\quad (mod\quad 7)\\ { 10 }^{ 4 }\equiv 4\quad (mod\quad 7)\\ { 10 }^{ 5 }\equiv 5\quad (mod\quad 7)$

Let us assume that the number of digits is at most 6. Then, the only way to make $x\equiv 0\quad (mod\quad 7)$ is by setting $x\equiv 3+6+5\equiv 14\equiv 0\quad (mod\quad 7)$ . Therefore, $x$ has '1' at the tens, thousands, and hundred thousands places.

$\therefore \quad x=101010$

Although I did not understand completely why the answer turned out to be 101010, I took a hint from the title. I just converted 42 into binary and got the answer right! Please can anyone explain why it actually happened? Meanwhile, I will make a few attempts to understand it myself.😇