Question

126n=86 find in base 2

Answer 1

To solve the equation \( 126n = 86 \) for \( n \), we first isolate \( n \):

\[
n = \frac{86}{126}
\]

Now, we can simplify the fraction. The greatest common divisor (GCD) of 86 and 126 is 2. Therefore, we can divide both the numerator and the denominator by 2:

\[
n = \frac{86 \div 2}{126 \div 2} = \frac{43}{63}
\]

This fraction \( \frac{43}{63} \) is the simplest form.

Next, we need to express this result in base 2. However, fractional values can be a bit complicated to work with in base 2. First, we can convert the numerator and the denominator to binary form separately.

### Step 1: Convert the numerator 43 to binary.

To convert 43 to binary, we divide by 2 and keep track of the remainders:

- \( 43 \div 2 = 21\) remainder \( 1 \)
- \( 21 \div 2 = 10 \) remainder \( 1 \)
- \( 10 \div 2 = 5 \) remainder \( 0 \)
- \( 5 \div 2 = 2 \) remainder \( 1 \)
- \( 2 \div 2 = 1 \) remainder \( 0 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)

Reading the remainders from bottom to top, \( 43 \) in binary is \( 101011 \).

### Step 2: Convert the denominator 63 to binary.

Now, for 63:

- \( 63 \div 2 = 31\) remainder \( 1 \)
- \( 31 \div 2 = 15 \) remainder \( 1 \)
- \( 15 \div 2 = 7 \) remainder \( 1 \)
- \( 7 \div 2 = 3 \) remainder \( 1 \)
- \( 3 \div 2 = 1 \) remainder \( 1 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)

Reading the remainders from bottom to top, \( 63 \) in binary is \( 111111 \).

### Step 3: Combine the results.

Thus, the fraction \( \frac{43}{63} \) in binary is represented as:

\[
\frac{101011}{111111}
\]

However, it's worth noting that this is a representation of the fraction itself and may not have a straightforward decimal-like representation in binary format.

In summary, \( n = \frac{43}{63} \) can be represented in binary as \( \frac{101011}{111111} \).