Divide 1101001$_{two}$ by 101$_{two}$

To solve this division problem in base two (binary), we use the long division method, similar to how we divide in base ten.

1. Analyze the question

We are dividing $1101001_2$ by $101_2$.

2. Explain the concept

In binary division, we look at the divisor ($101_2$) and see how many times it fits into the leading digits of the dividend ($1101001_2$). We use the same rules of binary subtraction:

• $0 - 0 = 0$
• $1 - 0 = 1$
• $1 - 1 = 0$
• $10 - 1 = 1$ (borrowing 1 from the next column)

3. Show the solution

Let's perform the long division:

• Step 1: How many times does $101$ go into $110$? It goes 1 time. • $1 \times 101 = 101$ • $110 - 101 = 001$
• Step 2: Bring down the next digit ($1$). Now we have $011$. $101$ goes into $011$ 0 times. • $0 \times 101 = 000$ • $011 - 000 = 011$
• Step 3: Bring down the next digit ($0$). Now we have $110$. $101$ goes into $110$ 1 time. • $1 \times 101 = 101$ • $110 - 101 = 001$
• Step 4: Bring down the final digit ($1$). Now we have $011$. $101$ goes into $011$ 0 times? Wait, let's re-check the alignment.

Let's re-calculate carefully: $1101001 \div 101$

1. $110 \div 101 = 1$ remainder $1$.
2. Bring down $1 \rightarrow 11$. $11 \div 101 = 0$.
3. Bring down $0 \rightarrow 110$. $110 \div 101 = 1$ remainder $1$.
4. Bring down $1 \rightarrow 11$. $11 \div 101 = 0$.

Actually, the standard long division for $1101001_2 \div 101_2$ is: $1101001_2 = 105_{10}$ $101_2 = 5_{10}$ $105 \div 5 = 21_{10}$

Now, convert $21_{10}$ back to binary: $21 \div 2 = 10$ rem $1$ $10 \div 2 = 5$ rem $0$ $5 \div 2 = 2$ rem $1$ $2 \div 2 = 1$ rem $0$ $1 \div 2 = 0$ rem $1$ Reading from bottom to top, we get $10101_2$.

4. Identify the answer

Comparing our result ($10101_2$) to the given options, it matches option D.

5. Tip

When you are stuck with binary arithmetic, convert the numbers to base ten (decimal), perform the operation (addition, subtraction, multiplication, or division), and then convert your final answer back to binary. It is often much faster and less prone to errors!