### Instruction: What is the base $2$ representation of $84_{10}$?### Response: To convert a base $10$ number to base $2$, I can use repeated division by $2$ and keep track of the remainders.
When I divide $84$ by $2$, I get a quotient of $42$ and a remainder of $0$.
I write down the remainder as the rightmost digit of the base $2$ representation, so it is $0$ so far.
Then I repeat the process with the quotient, $42$.
When I divide $42$ by $2$, I get a quotient of $21$ and a remainder of $0$.
I write down the remainder as the next digit to the left, so it is $00$ so far.
Then I repeat the process with the quotient, $21$.
When I divide $21$ by $2$, I get a quotient of $10$ and a remainder of $1$.
I write down the remainder as the next digit to the left, so it is $100$ so far.
Then I repeat the process with the quotient, $10$.
When I divide $10$ by $2$, I get a quotient of $5$ and a remainder of $0$.
I write down the remainder as the next digit to the left, so it is $0100$ so far.
Then I repeat the process with the quotient, $5$.
When I divide $5$ by $2$, I get a quotient of $2$ and a remainder of $1$.
I write down the remainder as the next digit to the left, so it is $10100$ so far.
Then I repeat the process with the quotient, $2$.
When I divide $2$ by $2$, I get a quotient of $1$ and a remainder of $0$.
I write down the remainder as the next digit to the left, so it is $010100$ so far.
Then I repeat the process with the quotient, $1$.
When I divide $1$ by $2$, I get a quotient of $0$ and a remainder of $1$.
I write down the remainder as the final digit to the left, so it is $1010100$.
This is the base $2$ representation of $84_{10}$.