Find $10110_2\times10100_2\div10_2$. Express your answer in base 2.
It is much easier to find $10110_2\div10_2$ and then multiply by $10100_2$ than it is to do the calculations in the original order. For $10110_2\div10_2$, since the last digit of $10110_2$ is a 0, we can simply take it off to get $1011_2$. This is similar to base 10, where $10110_{10}\div10_{10}=1011_{10}$. In base 2, each place represents a power of 2, and since we're dividing by 2, each place goes down by a power 2, so each digit shifts to the right. Now we find the product of $1011_2$ and $10100_2$. $$\begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c}
& & &1 &0 & 1 & 1_2 & & \\
& & & \times & 1& 0 & 1& 0 & 0_2 \\
\cline{1-9}& & &1 &0 &1 &1 & & \\
& & & & & &0 & & \\
& 1 &\stackrel{1}{0}&1 &1 &0 &0 &\downarrow &\downarrow \\
\cline{1-9}
&1 &1 &0 &1 &1 &1 &0 &0_2 \\
\end{array}$$The answer is $\boxed{11011100_2}$.
11011100_2
Level 5
Number Theory
Problem: Find $10110_2\times10100_2\div10_2$. Express your answer in base 2.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, first convert the binary numbers to decimal for easier calculations. 

Convert $10110_2$ to decimal:
1. $1 \times 2^4 = 16$
2. $0 \times 2^3 = 0$
3. $1 \times 2^2 = 4$
4. $1 \times 2^1 = 2$
5. $0 \times 2^0 = 0$
Adding these values gives $16 + 0 + 4 + 2 + 0 = 22$.

Convert $10100_2$ to decimal:
1. $1 \times 2^4 = 16$
2. $0 \times 2^3 = 0$
3. $1 \times 2^2 = 4$
4. $0 \times 2^1 = 0$
5. $0 \times 2^0 = 0$
Adding these values gives $16 + 0 + 4 + 0 + 0 = 20$.

Convert $10_2$ to decimal:
1. $1 \times 2^1 = 2$
2. $0 \times 2^0 = 0$
Adding these values gives $2$.

Now perform the multiplication and division in decimal:
1. Calculate $22 \times 20 = 440$.
2. Divide $440$ by $2$: $440 \div 2 = 220$.

Next, convert the decimal result back to binary:
1. Divide $220$ by $2$: quotient is $110$, remainder is $0$.
2. Divide $110$ by $2$: quotient is $55$, remainder is $0$.
3. Divide $55$ by $2$: quotient is $27$, remainder is $1$.
4. Divide $27$ by $2$: quotient is $13$, remainder is $1$.
5. Divide $13$ by $2$: quotient is $6$, remainder is $1$.
6. Divide $6$ by $2$: quotient is $3$, remainder is $0$.
7. Divide $3$ by $2$: quotient is $1$, remainder is $1$.
8. Divide $1$ by $2$: quotient is $0$, remainder is $1$.

Reading the remainders from bottom to top gives $11011100_2$. Thus, the final answer is $11011100_2$.<|END_OF_LATENT|> The final answer is $11011100_2$.