Solved on Feb 06, 2024

Convert 823 to base-5 notation. 823=five 823=\square_{\text{five}}

STEP 1

Assumptions
1. We want to convert the decimal number 823 to base-five.
2. In base-five, each digit represents powers of 5, starting with 505^0 on the right for the least significant digit and increasing powers of 5 as we move left.

STEP 2

Find the largest power of 5 that is less than or equal to 823. We can do this by trial and error or by using logarithms. For simplicity, we will use trial and error.

STEP 3

Start with 505^0 and keep increasing the power until the value is greater than 823.
50=15^0 = 1 51=55^1 = 5 52=255^2 = 25 53=1255^3 = 125 54=6255^4 = 625 55=31255^5 = 3125

STEP 4

We can see that 54=6255^4 = 625 is the largest power of 5 that is less than 823, and 555^5 is too large. So we will use 545^4 as the starting point.

STEP 5

Determine how many times 545^4 fits into 823 without exceeding it. This will give us the first digit in the base-five representation.
823÷54=823÷625823 \div 5^4 = 823 \div 625

STEP 6

Calculate the division to find the first digit.
823÷625=1 with a remainder of 198823 \div 625 = 1 \text{ with a remainder of } 198

STEP 7

The first digit (from the left) in the base-five representation is 1. Now we subtract 1×541 \times 5^4 from 823 to find the remainder that needs to be represented in lower powers of 5.
823(1×625)=198823 - (1 \times 625) = 198

STEP 8

Repeat the process for the next largest power of 5, which is 53=1255^3 = 125.
198÷53=198÷125198 \div 5^3 = 198 \div 125

STEP 9

Calculate the division to find the next digit.
198÷125=1 with a remainder of 73198 \div 125 = 1 \text{ with a remainder of } 73

STEP 10

The second digit (from the left) in the base-five representation is also 1. Subtract 1×531 \times 5^3 from 198 to find the next remainder.
198(1×125)=73198 - (1 \times 125) = 73

STEP 11

Repeat the process for the next largest power of 5, which is 52=255^2 = 25.
73÷52=73÷2573 \div 5^2 = 73 \div 25

STEP 12

Calculate the division to find the next digit.
73÷25=2 with a remainder of 2373 \div 25 = 2 \text{ with a remainder of } 23

STEP 13

The third digit (from the left) in the base-five representation is 2. Subtract 2×522 \times 5^2 from 73 to find the next remainder.
73(2×25)=2373 - (2 \times 25) = 23

STEP 14

Repeat the process for the next largest power of 5, which is 51=55^1 = 5.
23÷51=23÷523 \div 5^1 = 23 \div 5

STEP 15

Calculate the division to find the next digit.
23÷5=4 with a remainder of 323 \div 5 = 4 \text{ with a remainder of } 3

STEP 16

The fourth digit (from the left) in the base-five representation is 4. Subtract 4×514 \times 5^1 from 23 to find the next remainder.
23(4×5)=323 - (4 \times 5) = 3

STEP 17

Finally, the remainder 3 is less than 5 and represents the last digit in the base-five system, which is 50=15^0 = 1.

STEP 18

The fifth digit (from the left) in the base-five representation is 3.

STEP 19

Combine all the digits we found to get the number 823 in the base-five system.
823ten=11423five823_{\text{ten}} = 11423_{\text{five}}
The base-five representation of the decimal number 823 is 11423five11423_{\text{five}}.

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