Solved on Feb 06, 2024

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

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 $5^0$ 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 $5^0$ and keep increasing the power until the value is greater than 823.
$5^0 = 1$ $5^1 = 5$ $5^2 = 25$ $5^3 = 125$ $5^4 = 625$ $5^5 = 3125$

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

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

STEP 14

Repeat the process for the next largest power of 5, which is $5^1 = 5$ .
$23 \div 5^1 = 23 \div 5$

STEP 15

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

STEP 16

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

STEP 17

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

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.
$823_{\text{ten}} = 11423_{\text{five}}$
The base-five representation of the decimal number 823 is $11423_{\text{five}}$ .

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

STEP 1

Assumptions1. We want to convert the decimal number 823 to base-five.2. In base-five, each digit represents powers of 5, starting with 505^050 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 505^050 and keep increasing the power until the value is greater than 823.50=15^0 = 150=1 51=55^1 = 551=5 52=255^2 = 2552=25 53=1255^3 = 12553=125 54=6255^4 = 62554=625 55=31255^5 = 312555=3125

STEP 4

We can see that 54=6255^4 = 62554=625 is the largest power of 5 that is less than 823, and 555^555 is too large. So we will use 545^454 as the starting point.

STEP 5

Determine how many times 545^454 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 625823÷54=823÷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 } 198823÷625=1 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 5^41×54 from 823 to find the remainder that needs to be represented in lower powers of 5.823−(1×625)=198823 - (1 \times 625) = 198823−(1×625)=198

STEP 8

Repeat the process for the next largest power of 5, which is 53=1255^3 = 12553=125.198÷53=198÷125198 \div 5^3 = 198 \div 125198÷53=198÷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 } 73198÷125=1 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 5^31×53 from 198 to find the next remainder.198−(1×125)=73198 - (1 \times 125) = 73198−(1×125)=73

STEP 11

Repeat the process for the next largest power of 5, which is 52=255^2 = 2552=25.73÷52=73÷2573 \div 5^2 = 73 \div 2573÷52=73÷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 } 2373÷25=2 with a remainder of 23

STEP 13

The third digit (from the left) in the base-five representation is 2. Subtract 2×522 \times 5^22×52 from 73 to find the next remainder.73−(2×25)=2373 - (2 \times 25) = 2373−(2×25)=23

STEP 14

Repeat the process for the next largest power of 5, which is 51=55^1 = 551=5.23÷51=23÷523 \div 5^1 = 23 \div 523÷51=23÷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 } 323÷5=4 with a remainder of 3

STEP 16

The fourth digit (from the left) in the base-five representation is 4. Subtract 4×514 \times 5^14×51 from 23 to find the next remainder.23−(4×5)=323 - (4 \times 5) = 323−(4×5)=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 = 150=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}}823ten​=11423five​The base-five representation of the decimal number 823 is 11423five11423_{\text{five}}11423five​.

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

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 $5^0$ on the right for the least significant digit and increasing powers of 5 as we move left.

Start with $5^0$ and keep increasing the power until the value is greater than 823.
$5^0 = 1$ $5^1 = 5$ $5^2 = 25$ $5^3 = 125$ $5^4 = 625$ $5^5 = 3125$

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

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

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

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

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

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

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

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

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

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

Repeat the process for the next largest power of 5, which is $5^1 = 5$ .
$23 \div 5^1 = 23 \div 5$

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

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

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

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