QuestionCalculate the following in the specified bases: a. $31_{\text{five}} \cdot 3_{\text{five}}$ b. $31_{\text{five}} \div 3_{\text{five}}$ c. $32_{\text{six}} \cdot 23_{\text{six}}$

Studdy Solution

STEP 1

Assumptions1. We are performing arithmetic operations in different bases. . The base of the numbers is specified in the subscript.
3. We need to perform the operations and provide the answers in the same base as the input numbers.

STEP 2

First, let's solve the multiplication operation in base five. We have $31_{\text {five }} \cdot_{\text {five }}$ .

STEP 3

To perform this operation, we first convert the numbers to base ten, perform the multiplication, and then convert the result back to base five.

STEP 4

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
$31_{\text {five }} =3 \cdot^1 +1 \cdot^0 =15 +1 =16_{\text {ten }}$ $3_{\text {five }} =3_{\text {ten }}$

STEP 5

Perform the multiplication in base ten.
$16_{\text {ten }} \cdot3_{\text {ten }} =48_{\text {ten }}$

STEP 6

Convert the result back to base five. $48_{\text {ten }} =1 \cdot5^3 +4 \cdot5^1 +3 \cdot5^0 =143_{\text {five }}$ So, $31_{\text {five }} \cdot3_{\text {five }} =143_{\text {five }}$ .

STEP 7

Next, let's solve the division operation in base five. We have $31_{\text {five }} \div3_{\text {five }}$ .

STEP 8

To perform this operation, we first convert the numbers to base ten, perform the division, and then convert the result back to base five.

STEP 9

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
We already converted these numbers in previous steps. So, $31_{\text {five }} =16_{\text {ten }}$ and $3_{\text {five }} =3_{\text {ten }}$ .

STEP 10

Perform the division in base ten.
$16_{\text {ten }} \div3_{\text {ten }} =5_{\text {ten }} \text{ remainder }_{\text {ten }}$

STEP 11

Convert the quotient and remainder back to base five.
$5_{\text {ten }} = \cdot5^ =10_{\text {five }}$ $_{\text {ten }} =_{\text {five }}$ So, $31_{\text {five }} \div3_{\text {five }} =10_{\text {five }}$ remainder $_{\text {five }}$ .

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QuestionCalculate the following in the specified bases: a. $31_{\text{five}} \cdot 3_{\text{five}}$ b. $31_{\text{five}} \div 3_{\text{five}}$ c. $32_{\text{six}} \cdot 23_{\text{six}}$

Studdy Solution

STEP 1

Assumptions1. We are performing arithmetic operations in different bases. . The base of the numbers is specified in the subscript.
3. We need to perform the operations and provide the answers in the same base as the input numbers.

STEP 2

First, let's solve the multiplication operation in base five. We have $31_{\text {five }} \cdot_{\text {five }}$ .

STEP 3

To perform this operation, we first convert the numbers to base ten, perform the multiplication, and then convert the result back to base five.

STEP 4

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
$31_{\text {five }} =3 \cdot^1 +1 \cdot^0 =15 +1 =16_{\text {ten }}$ $3_{\text {five }} =3_{\text {ten }}$

STEP 5

Perform the multiplication in base ten.
$16_{\text {ten }} \cdot3_{\text {ten }} =48_{\text {ten }}$

STEP 6

Convert the result back to base five. $48_{\text {ten }} =1 \cdot5^3 +4 \cdot5^1 +3 \cdot5^0 =143_{\text {five }}$ So, $31_{\text {five }} \cdot3_{\text {five }} =143_{\text {five }}$ .

STEP 7

Next, let's solve the division operation in base five. We have $31_{\text {five }} \div3_{\text {five }}$ .

STEP 8

To perform this operation, we first convert the numbers to base ten, perform the division, and then convert the result back to base five.

STEP 9

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
We already converted these numbers in previous steps. So, $31_{\text {five }} =16_{\text {ten }}$ and $3_{\text {five }} =3_{\text {ten }}$ .

STEP 10

Perform the division in base ten.
$16_{\text {ten }} \div3_{\text {ten }} =5_{\text {ten }} \text{ remainder }_{\text {ten }}$

STEP 11

Convert the quotient and remainder back to base five.
$5_{\text {ten }} = \cdot5^ =10_{\text {five }}$ $_{\text {ten }} =_{\text {five }}$ So, $31_{\text {five }} \div3_{\text {five }} =10_{\text {five }}$ remainder $_{\text {five }}$ .

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QuestionCalculate the following in the specified bases: a. 31five⋅3five31_{\text{five}} \cdot 3_{\text{five}}31five​⋅3five​ b. 31five÷3five31_{\text{five}} \div 3_{\text{five}}31five​÷3five​ c. 32six⋅23six32_{\text{six}} \cdot 23_{\text{six}}32six​⋅23six​

Studdy Solution

STEP 1

Assumptions1. We are performing arithmetic operations in different bases. . The base of the numbers is specified in the subscript.3. We need to perform the operations and provide the answers in the same base as the input numbers.

STEP 2

First, let's solve the multiplication operation in base five. We have 31five ⋅five 31_{\text {five }} \cdot_{\text {five }}31five ​⋅five ​.

STEP 3

To perform this operation, we first convert the numbers to base ten, perform the multiplication, and then convert the result back to base five.

STEP 4

Convert 31five 31_{\text {five }}31five ​ and 3five 3_{\text {five }}3five ​ to base ten.31five =3⋅1+1⋅0=15+1=16ten 31_{\text {five }} =3 \cdot^1 +1 \cdot^0 =15 +1 =16_{\text {ten }}31five ​=3⋅1+1⋅0=15+1=16ten ​3five =3ten 3_{\text {five }} =3_{\text {ten }}3five ​=3ten ​

STEP 5

Perform the multiplication in base ten.16ten ⋅3ten =48ten 16_{\text {ten }} \cdot3_{\text {ten }} =48_{\text {ten }}16ten ​⋅3ten ​=48ten ​

STEP 6

STEP 7

Next, let's solve the division operation in base five. We have 31five ÷3five 31_{\text {five }} \div3_{\text {five }}31five ​÷3five ​.

STEP 8

To perform this operation, we first convert the numbers to base ten, perform the division, and then convert the result back to base five.

STEP 9

Convert 31five 31_{\text {five }}31five ​ and 3five 3_{\text {five }}3five ​ to base ten.We already converted these numbers in previous steps. So, 31five =16ten 31_{\text {five }} =16_{\text {ten }}31five ​=16ten ​ and 3five =3ten 3_{\text {five }} =3_{\text {ten }}3five ​=3ten ​.

STEP 10

Perform the division in base ten.16ten ÷3ten =5ten remainder ten 16_{\text {ten }} \div3_{\text {ten }} =5_{\text {ten }} \text{ remainder }_{\text {ten }}16ten ​÷3ten ​=5ten ​ remainder ten ​

STEP 11

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QuestionCalculate the following in the specified bases: a. $31_{\text{five}} \cdot 3_{\text{five}}$ b. $31_{\text{five}} \div 3_{\text{five}}$ c. $32_{\text{six}} \cdot 23_{\text{six}}$

Assumptions1. We are performing arithmetic operations in different bases. . The base of the numbers is specified in the subscript.
3. We need to perform the operations and provide the answers in the same base as the input numbers.

First, let's solve the multiplication operation in base five. We have $31_{\text {five }} \cdot_{\text {five }}$ .

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
$31_{\text {five }} =3 \cdot^1 +1 \cdot^0 =15 +1 =16_{\text {ten }}$ $3_{\text {five }} =3_{\text {ten }}$

Perform the multiplication in base ten.
$16_{\text {ten }} \cdot3_{\text {ten }} =48_{\text {ten }}$

Next, let's solve the division operation in base five. We have $31_{\text {five }} \div3_{\text {five }}$ .

Convert $31_{\text {five }}$ and $3_{\text {five }}$ to base ten.
We already converted these numbers in previous steps. So, $31_{\text {five }} =16_{\text {ten }}$ and $3_{\text {five }} =3_{\text {ten }}$ .

Perform the division in base ten.
$16_{\text {ten }} \div3_{\text {ten }} =5_{\text {ten }} \text{ remainder }_{\text {ten }}$