Answer :

To convert the decimal number [tex]$115$[/tex] to base-five, we perform successive divisions by [tex]$5$[/tex] and keep track of the remainders. Each remainder represents a digit in the base-five number, starting from the least significant digit.

Step 1: Divide [tex]$115$[/tex] by [tex]$5$[/tex].
We have
[tex]$$115 \div 5 = 23 \quad \text{with a remainder of } 0.$$[/tex]
Thus, the least significant digit of the base-five number is [tex]$0$[/tex].

Step 2: Now, take the quotient from the previous division ([tex]$23$[/tex]) and divide it by [tex]$5$[/tex].
We get
[tex]$$23 \div 5 = 4 \quad \text{with a remainder of } 3.$$[/tex]
So, the next digit is [tex]$3$[/tex].

Step 3: Continue with the new quotient ([tex]$4$[/tex]) and divide it by [tex]$5$[/tex].
Since
[tex]$$4 \div 5 = 0 \quad \text{with a remainder of } 4,$$[/tex]
the process stops here because the quotient is now [tex]$0$[/tex]. This remainder, [tex]$4$[/tex], is the most significant digit.

Final Step: Write the digits from the last remainder to the first. In order, they are:
[tex]$$4 \quad 3 \quad 0.$$[/tex]

Thus, the number [tex]$115$[/tex] in base-ten is written as
[tex]$$115 = 430_{\text{five}}.$$[/tex]

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