Convert `(35)_(10)` into base 3.

Answer

To convert the decimal number \(35\) (base \(10\)) into base \(3\), we can follow these steps:

### Step 1: Divide the number by the new base
We start by dividing \(35\) by \(3\).

\[ 35 \div 3 = 11 \quad \text{(Quotient)} \quad \text{Remainder: } 2 \]

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

### Step 3: Repeat the process with the quotient
Next, we take the quotient \(11\) and divide it by \(3\).

\[ 11 \div 3 = 3 \quad \text{(Quotient)} \quad \text{Remainder: } 2 \]

### Step 4: Record the new remainder
The remainder from this division is also \(2\). We keep this as well.

### Step 5: Continue dividing the new quotient
Now, we take the quotient \(3\) and divide it by \(3\).

\[ 3 \div 3 = 1 \quad \text{(Quotient)} \quad \text{Remainder: } 0 \]

### Step 6: Record the remainder
The remainder from this division is \(0\). We will keep this too.

### Step 7: Divide the last quotient
Finally, we take the quotient \(1\) and divide it by \(3\).

\[ 1 \div 3 = 0 \quad \text{(Quotient)} \quad \text{Remainder: } 1 \]

### Step 8: Record the last remainder
The remainder from this division is \(1\).

### Step 9: Compile the remainders
Now we have all the remainders from our divisions:
- From \(1\): \(1\)
- From \(3\): \(0\)
- From \(11\): \(2\)
- From \(35\): \(2\)

### Step 10: Write the base 3 number
We write the remainders in reverse order (from last to first):
Thus, \(35\) in base \(3\) is \(1022\).

### Final Answer
\[ (35)_{10} = (1022)_{3} \]

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