Convert `(127)_(10)` into binary system.

Answer

To convert the decimal number \(127\) (base \(10\)) into binary (base \(2\)), we will follow these steps:

### Step 1: Divide the number by \(2\)
We start by dividing \(127\) by \(2\).

\[ 127 \div 2 = 63 \quad \text{Remainder} = 1 \]

### Step 2: Record the remainder
The remainder from this division is \(1\). We will keep this remainder for our binary representation.

### Step 3: Repeat the division
Now we take the quotient \(63\) and divide it by \(2\).

\[ 63 \div 2 = 31 \quad \text{Remainder} = 1 \]

### Step 4: Record the remainder
The remainder is again \(1\). We will keep this as well.

### Step 5: Continue dividing
Next, we take \(31\) and divide it by \(2\).

\[ 31 \div 2 = 15 \quad \text{Remainder} = 1 \]

### Step 6: Record the remainder
The remainder is \(1\). We will note this down.

### Step 7: Divide \(15\) by \(2\)
Now, we divide \(15\) by \(2\).

\[ 15 \div 2 = 7 \quad \text{Remainder} = 1 \]

### Step 8: Record the remainder
The remainder is \(1\). We will keep this as well.

### Step 9: Divide \(7\) by \(2\)
Next, we divide \(7\) by \(2\).

\[ 7 \div 2 = 3 \quad \text{Remainder} = 1 \]

### Step 10: Record the remainder
The remainder is \(1\). We will note this down.

### Step 11: Divide \(3\) by \(2\)
Now we divide \(3\) by \(2\).

\[ 3 \div 2 = 1 \quad \text{Remainder} = 1 \]

### Step 12: Record the remainder
The remainder is \(1\). We will keep this as well.

### Step 13: Divide \(1\) by \(2\)
Finally, we divide \(1\) by \(2\).

\[ 1 \div 2 = 0 \quad \text{Remainder} = 1 \]

### Step 14: Record the remainder
The remainder is \(1\). We will note this down.

### Step 15: Collect all remainders
Now we have all the remainders from the divisions:
- From bottom to top, the remainders are:
- \(1\)
- \(1\)
- \(1\)
- \(1\)
- \(1\)
- \(1\)
- \(1\)

### Step 16: Write the binary representation
Putting it all together, we read the remainders from bottom to top to get the binary representation of \(127\):

\[ (1111111)_2 \]

### Final Answer
Thus, the binary representation of \(127\) in base \(10\) is:

\[ (1111111)_2 \]

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