Answer :
Sure, let's go through the process of converting each given base ten number to base six step-by-step.
### a. Convert 12 from base ten to base six:
To convert 12 to base six, we divide 12 by 6 and record the remainders:
1. [tex]\( 12 ÷ 6 = 2 \)[/tex] remainder [tex]\( 0 \)[/tex] (first digit from the left)
2. So, 12 in base ten is [tex]\( 20 \)[/tex] in base six.
### b. Convert 31 from base ten to base six:
To convert 31 to base six, we divide 31 by 6 and record the remainders:
1. [tex]\( 31 ÷ 6 = 5 \)[/tex] remainder [tex]\( 1 \)[/tex]
2. [tex]\( 5 ÷ 6 = 0 \)[/tex] remainder [tex]\( 5 \)[/tex] (first digit from the left)
3. So, 31 in base ten is [tex]\( 51 \)[/tex] in base six.
### c. Convert 49 from base ten to base six:
To convert 49 to base six, we divide 49 by 6 and record the remainders:
1. [tex]\( 49 ÷ 6 = 8 \)[/tex] remainder [tex]\( 1 \)[/tex]
2. [tex]\( 8 ÷ 6 = 1 \)[/tex] remainder [tex]\( 2 \)[/tex]
3. [tex]\( 1 ÷ 6 = 0 \)[/tex] remainder [tex]\( 1 \)[/tex] (first digit from the left)
4. So, 49 in base ten is [tex]\( 121 \)[/tex] in base six.
### d. Convert 68 from base ten to base six:
To convert 68 to base six, we divide 68 by 6 and record the remainders:
1. [tex]\( 68 ÷ 6 = 11 \)[/tex] remainder [tex]\( 2 \)[/tex]
2. [tex]\( 11 ÷ 6 = 1 \)[/tex] remainder [tex]\( 5 \)[/tex]
3. [tex]\( 1 ÷ 6 = 0 \)[/tex] remainder [tex]\( 1 \)[/tex] (first digit from the left)
4. So, 68 in base ten is [tex]\( 152 \)[/tex] in base six.
### e. Convert 725 from base ten to base six:
To convert 725 to base six, we divide 725 by 6 and record the remainders:
1. [tex]\( 725 ÷ 6 = 120 \)[/tex] remainder [tex]\( 5 \)[/tex]
2. [tex]\( 120 ÷ 6 = 20 \)[/tex] remainder [tex]\( 0 \)[/tex]
3. [tex]\( 20 ÷ 6 = 3 \)[/tex] remainder [tex]\( 2 \)[/tex]
4. [tex]\( 3 ÷ 6 = 0 \)[/tex] remainder [tex]\( 3 \)[/tex] (first digit from the left)
5. So, 725 in base ten is [tex]\( 3205 \)[/tex] in base six.
### f. Convert 1000 from base ten to base six:
To convert 1000 to base six, we divide 1000 by 6 and record the remainders:
1. [tex]\( 1000 ÷ 6 = 166 \)[/tex] remainder [tex]\( 4 \)[/tex]
2. [tex]\( 166 ÷ 6 = 27 \)[/tex] remainder [tex]\( 4 \)[/tex]
3. [tex]\( 27 ÷ 6 = 4 \)[/tex] remainder [tex]\( 3 \)[/tex]
4. [tex]\( 4 ÷ 6 = 0 \)[/tex] remainder [tex]\( 4 \)[/tex] (first digit from the left)
5. So, 1000 in base ten is [tex]\( 4344 \)[/tex] in base six.
So the conversions of the given base ten numbers to base six are:
a. 12 in base ten is [tex]\( 20 \)[/tex] in base six.
b. 31 in base ten is [tex]\( 51 \)[/tex] in base six.
c. 49 in base ten is [tex]\( 121 \)[/tex] in base six.
d. 68 in base ten is [tex]\( 152 \)[/tex] in base six.
e. 725 in base ten is [tex]\( 3205 \)[/tex] in base six.
f. 1000 in base ten is [tex]\( 4344 \)[/tex] in base six.
### a. Convert 12 from base ten to base six:
To convert 12 to base six, we divide 12 by 6 and record the remainders:
1. [tex]\( 12 ÷ 6 = 2 \)[/tex] remainder [tex]\( 0 \)[/tex] (first digit from the left)
2. So, 12 in base ten is [tex]\( 20 \)[/tex] in base six.
### b. Convert 31 from base ten to base six:
To convert 31 to base six, we divide 31 by 6 and record the remainders:
1. [tex]\( 31 ÷ 6 = 5 \)[/tex] remainder [tex]\( 1 \)[/tex]
2. [tex]\( 5 ÷ 6 = 0 \)[/tex] remainder [tex]\( 5 \)[/tex] (first digit from the left)
3. So, 31 in base ten is [tex]\( 51 \)[/tex] in base six.
### c. Convert 49 from base ten to base six:
To convert 49 to base six, we divide 49 by 6 and record the remainders:
1. [tex]\( 49 ÷ 6 = 8 \)[/tex] remainder [tex]\( 1 \)[/tex]
2. [tex]\( 8 ÷ 6 = 1 \)[/tex] remainder [tex]\( 2 \)[/tex]
3. [tex]\( 1 ÷ 6 = 0 \)[/tex] remainder [tex]\( 1 \)[/tex] (first digit from the left)
4. So, 49 in base ten is [tex]\( 121 \)[/tex] in base six.
### d. Convert 68 from base ten to base six:
To convert 68 to base six, we divide 68 by 6 and record the remainders:
1. [tex]\( 68 ÷ 6 = 11 \)[/tex] remainder [tex]\( 2 \)[/tex]
2. [tex]\( 11 ÷ 6 = 1 \)[/tex] remainder [tex]\( 5 \)[/tex]
3. [tex]\( 1 ÷ 6 = 0 \)[/tex] remainder [tex]\( 1 \)[/tex] (first digit from the left)
4. So, 68 in base ten is [tex]\( 152 \)[/tex] in base six.
### e. Convert 725 from base ten to base six:
To convert 725 to base six, we divide 725 by 6 and record the remainders:
1. [tex]\( 725 ÷ 6 = 120 \)[/tex] remainder [tex]\( 5 \)[/tex]
2. [tex]\( 120 ÷ 6 = 20 \)[/tex] remainder [tex]\( 0 \)[/tex]
3. [tex]\( 20 ÷ 6 = 3 \)[/tex] remainder [tex]\( 2 \)[/tex]
4. [tex]\( 3 ÷ 6 = 0 \)[/tex] remainder [tex]\( 3 \)[/tex] (first digit from the left)
5. So, 725 in base ten is [tex]\( 3205 \)[/tex] in base six.
### f. Convert 1000 from base ten to base six:
To convert 1000 to base six, we divide 1000 by 6 and record the remainders:
1. [tex]\( 1000 ÷ 6 = 166 \)[/tex] remainder [tex]\( 4 \)[/tex]
2. [tex]\( 166 ÷ 6 = 27 \)[/tex] remainder [tex]\( 4 \)[/tex]
3. [tex]\( 27 ÷ 6 = 4 \)[/tex] remainder [tex]\( 3 \)[/tex]
4. [tex]\( 4 ÷ 6 = 0 \)[/tex] remainder [tex]\( 4 \)[/tex] (first digit from the left)
5. So, 1000 in base ten is [tex]\( 4344 \)[/tex] in base six.
So the conversions of the given base ten numbers to base six are:
a. 12 in base ten is [tex]\( 20 \)[/tex] in base six.
b. 31 in base ten is [tex]\( 51 \)[/tex] in base six.
c. 49 in base ten is [tex]\( 121 \)[/tex] in base six.
d. 68 in base ten is [tex]\( 152 \)[/tex] in base six.
e. 725 in base ten is [tex]\( 3205 \)[/tex] in base six.
f. 1000 in base ten is [tex]\( 4344 \)[/tex] in base six.
