2. Determine the base- five representations for each of the given numbers. Remember that a numeral with no subscript is understood to be n base ten.
a.47
b. 179
c. 354
Divide the number by the base and note the remainder on the right.
The digits from bottom to top is the number required.
Example:
Find 37 to base 5.
37 R 2 (37/5=7 R2)
7 R 2 (7/5=1 R2)
1
So the number is 1225
To determine the base-5 representation of a number, you need to divide the number by the base (which in this case is 5), and keep track of the remainders at each step until the quotient becomes zero. The remainders, read in reverse order, will give you the base-5 representation of the number.
Let's apply this process to each of the given numbers:
a. 47:
Divide 47 by 5:
47 ÷ 5 = 9 remainder 2
So, the first remainder is 2.
Divide 9 by 5:
9 ÷ 5 = 1 remainder 4
The second remainder is 4.
Divide 1 by 5:
1 ÷ 5 = 0 remainder 1
The third and final remainder is 1.
Now, let's read the remainders in reverse order: 1, 4, 2.
Therefore, the base-5 representation of 47 is 142.
b. 179:
Divide 179 by 5:
179 ÷ 5 = 35 remainder 4
The first remainder is 4.
Divide 35 by 5:
35 ÷ 5 = 7 remainder 0
The second remainder is 0.
Divide 7 by 5:
7 ÷ 5 = 1 remainder 2
The third remainder is 2.
Divide 1 by 5:
1 ÷ 5 = 0 remainder 1
The fourth and final remainder is 1.
Read the remainders in reverse order: 1, 2, 0, 4.
Therefore, the base-5 representation of 179 is 1204.
c. 354:
Divide 354 by 5:
354 ÷ 5 = 70 remainder 4
The first remainder is 4.
Divide 70 by 5:
70 ÷ 5 = 14 remainder 0
The second remainder is 0.
Divide 14 by 5:
14 ÷ 5 = 2 remainder 4
The third remainder is 4.
Divide 2 by 5:
2 ÷ 5 = 0 remainder 2
The fourth and final remainder is 2.
Read the remainders in reverse order: 2, 4, 0, 4.
Therefore, the base-5 representation of 354 is 2404.