# Chapter 4 - Number Representation and Calculation - 4.2 Number Bases in Positional Systems - Exercise Set 4.2: 58

$123_{\text{five}}$

#### Work Step by Step

To convert a numeral in base eight to a numeral in base five, the following steps must be performed: (1) Convert the numeral in base eight to base ten. $46_{\text{eight}}=(4\times 8^1) + (6 \times 1)=32+6=38$ (2) Convert the result in Step (1) to a numeral in base five using division. The place values in base five are: $5^2, 5^1, 1$. The place values less than or equal to 38 are: $5^2, 5^1,$ and $1$. Divide $38$ by $5^2$ or $25$: $38 \div 25 = 1$ remainder $13$ Divide $13$ by $5^1$ or $5$: $13 \div 5 = 2$ remainder $3$ Divide $3$ by $1$: $3 \div 1 =3$ Thus, $38 = (1 \times 5^2) + (2 \times 5^1) + (3 \times 1) \\38= 123_{\text{five}}$

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