Sure, let's solve the problem of dividing [tex]\(8612_{12}\)[/tex] by [tex]\(3E_{12}\)[/tex] step-by-step.
### Step 1: Converting from Base 12 to Decimal
First, we need to convert the numbers from base twelve to decimal (base ten).
#### Converting [tex]\(8612_{12}\)[/tex] to Decimal:
Each digit in the number represents a power of 12:
[tex]\[8612_{12} = 8 \times 12^3 + 6 \times 12^2 + 1 \times 12^1 + 2 \times 12^0\][/tex]
Calculate each term:
[tex]\[ 8 \times 12^3 = 8 \times 1728 = 13824 \][/tex]
[tex]\[ 6 \times 12^2 = 6 \times 144 = 864 \][/tex]
[tex]\[ 1 \times 12^1 = 1 \times 12 = 12 \][/tex]
[tex]\[ 2 \times 12^0 = 2 \times 1 = 2 \][/tex]
Now, add them up:
[tex]\[ 8612_{12} = 13824 + 864 + 12 + 2 = 14702_{10} \][/tex]
#### Converting [tex]\(3E_{12}\)[/tex] to Decimal:
Here, [tex]\(E\)[/tex] represents 11 in decimal:
[tex]\[3E_{12} = 3 \times 12^1 + 11 \times 12^0\][/tex]
Calculate each term:
[tex]\[ 3 \times 12^1 = 3 \times 12 = 36 \][/tex]
[tex]\[ 11 \times 12^0 = 11 \times 1 = 11 \][/tex]
Now, add them up:
[tex]\[ 3E_{12} = 36 + 11 = 47_{10} \][/tex]
### Step 2: Perform the Division in Decimal
Next, we need to perform the division with the decimal equivalents:
[tex]\[ 14702_{10} \div 47_{10} \][/tex]
Perform the division:
[tex]\[ 14702 \div 47 = 312 \][/tex]
### Step 3: Convert the Result Back to Base 12
Finally, we need to convert the quotient [tex]\(312_{10}\)[/tex] back to base twelve.
#### Converting [tex]\(312_{10}\)[/tex] to Base 12:
To convert from decimal to base twelve, repeatedly divide the number by 12 and keep track of the remainders.
1. [tex]\(312 \div 12 = 26\)[/tex] remainder [tex]\(0\)[/tex]
2. [tex]\(26 \div 12 = 2\)[/tex] remainder [tex]\(2\)[/tex]
3. [tex]\(2 \div 12 = 0\)[/tex] remainder [tex]\(2\)[/tex]
Reading the remainders from bottom to top, we get:
[tex]\[ 312_{10} = 220_{12} \][/tex]
Therefore, the result of the division [tex]\( 8612_{12} \div 3E_{12} \)[/tex] is [tex]\( 220_{12} \)[/tex].
