Questions: Divide in the indicated base. 3 5 longdiv 2 3 0 5 The quotient is 5 R 5.

Transcript text: Divide in the indicated base. \[ 3 _ { 5 } \longdiv { 2 3 0 _ { 5 } } \] The quotient is $\square$ ${ }_{5}$ R $\square$ 5.

Solution

Step 1: Convert from Base 5 to Base 10

We start by converting the numbers from base 5 to base 10. The number $230_5$ is calculated as follows:

\[ 230_5 = 2 \cdot 5^2 + 3 \cdot 5^1 + 0 \cdot 5^0 = 2 \cdot 25 + 3 \cdot 5 + 0 = 50 + 15 + 0 = 65 \]

The number $3_5$ is simply:

\[ 3_5 = 3 \cdot 5^0 = 3 \]

Step 2: Perform the Division in Base 10

Next, we perform the division of the two base 10 numbers:

\[ \frac{65}{3} = 21 \quad \text{with a remainder of} \quad 2 \]

Step 3: Convert the Quotient and Remainder Back to Base 5

Now, we convert the quotient $21$ and the remainder $2$ back to base 5.

For the quotient $21$:

\[ 21_{10} = 4 \cdot 5^1 + 1 \cdot 5^0 = 41_5 \]

For the remainder $2$:

\[ 2_{10} = 2 \cdot 5^0 = 2_5 \]

The quotient is $41_5$ and the remainder is $2_5$. Thus, the final answer is:

\[ \boxed{41_5 \text{ R } 2_5} \]

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