How do you multiply 4213 base 5 by 23 base 5?

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to solve the multiplication problem in base 5:

Set up the multiplication.
$\begin{array}{c} & 4 & 2 & 1 & 3_5 \\ \times & & & 2 & 3_5 \\ \hline \end{array}$
Multiply $4213_5$ by $3_5$ .
- $3_5 \times 3_5 = 9_{10} = 14_5$ . Write down 4, carry over 1.
- $3_5 \times 1_5 + 1_5 (\text{carry}) = 3_{10} + 1_{10} = 4_{10} = 4_5$ . Write down 4.
- $3_5 \times 2_5 = 6_{10} = 11_5$ . Write down 1, carry over 1.
- $3_5 \times 4_5 + 1_5 (\text{carry}) = 12_{10} + 1_{10} = 13_{10} = 23_5$ . Write down 23. The first partial product is $23144_5$ .
Multiply $4213_5$ by $2_5$ (shifted one place to the left).
- $2_5 \times 3_5 = 6_{10} = 11_5$ . Write down 1, carry over 1.
- $2_5 \times 1_5 + 1_5 (\text{carry}) = 2_{10} + 1_{10} = 3_{10} = 3_5$ . Write down 3.
- $2_5 \times 2_5 = 4_{10} = 4_5$ . Write down 4.
- $2_5 \times 4_5 = 8_{10} = 13_5$ . Write down 13. The second partial product is $134310_5$ .
Add the partial products in base 5.
$\begin{array}{c} & & 2 & 3 & 1 & 4 & 4_5 \\ + & 1 & 3 & 4 & 3 & 1 & 0_5 \\ \hline & 1 & 1 & 3 & 0 & 0 & 4_5 \\ \end{array}$
- $4+0 = 4_5$
- $4+1 = 5_{10} = 10_5$ . Write 0, carry 1.
- $1+3+1 (\text{carry}) = 5_{10} = 10_5$ . Write 0, carry 1.
- $3+4+1 (\text{carry}) = 8_{10} = 13_5$ . Write 3, carry 1.
- $2+3+1 (\text{carry}) = 6_{10} = 11_5$ . Write 1, carry 1.
- $0+1 (\text{carry}) = 1_5$ . Write 1.
The final answer is:
$4213_5 \times 23_5 = \boxed{113004_5}$

Answer