### Instruction: What is the base $2$ representation of $84_{10}$?### Response: I need to convert the base $10$ number $84$ to base $2$, which means I need to express it as a sum of powers of $2$.
I can start by finding the largest power of $2$ that is less than or equal to $84$.
That is $2^6 = 64$.
I can subtract $64$ from $84$ to get the remainder: $84 - 64 = 20$.
I can repeat the process with $20$, finding the largest power of $2$ that is less than or equal to it.
That is $2^4 = 16$.
I can subtract $16$ from $20$ to get the remainder: $20 - 16 = 4$.
I can repeat the process with $4$, finding the largest power of $2$ that is less than or equal to it.
That is $2^2 = 4$.
I can subtract $4$ from $4$ to get the remainder: $4 - 4 = 0$.
I have no more remainder left, so I can stop the process.
To write the base $2$ representation, I can use a binary digit for each power of $2$ that I used in the sum, starting from the highest power and going down to the lowest power.
For each power of $2$ that I used, I write a $1$ in the corresponding position.
For each power of $2$ that I did not use, I write a $0$ in the corresponding position.
The highest power of $2$ that I used was $2^6$, so the first digit is $1$.
The next power of $2$ is $2^5$, which I did not use, so the second digit is $0$.
The next power of $2$ is $2^4$, which I used, so the third digit is $1$.
The next power of $2$ is $2^3$, which I did not use, so the fourth digit is $0$.
The next power of $2$ is $2^2$, which I used, so the fifth digit is $1$.
The next power of $2$ is $2^1$, which I did not use, so the sixth digit is $0$.
The last power of $2$ is $2^0$, which I did not use, so the seventh digit is $0$.