To convert the decimal number 137 to base 5, we need to express it as a sum of powers of 5. This means we want to find digits \( a, b, c, d \) such that:
\[ 137 = a \times 5^3 + b \times 5^2 + c \times 5^1 + d \times 5^0 \] where \( a, b, c, d \) are digits from 0 to 4.
Step 1: Find the largest power of 5 less than or equal to 137.
The powers of 5 are:
- \( 5^0 = 1 \)
- \( 5^1 = 5 \)
- \( 5^2 = 25 \)
- \( 5^3 = 125 \)
- \( 5^4 = 625 \) (which is too large)
The largest useful power is \( 125 \), or \( 5^3 \).
Step 2: Divide 137 by 125 to find the coefficient for \( 5^3 \):
\[ a = \left\lfloor \frac{137}{125} \right\rfloor = 1 \] So, \( 1 \times 125 = 125 \).
Step 3: Subtract and repeat for the next lower power:
\[ 137 - 125 = 12 \] Now, divide 12 by \( 25 \) (next lower power): \[ b = \left\lfloor \frac{12}{25} \right\rfloor = 0 \] So, \( 0 \times 25 = 0 \).
Step 4: Move to \( 5^1 \):
Now, divide 12 by 5: \[ c = \left\lfloor \frac{12}{5} \right\rfloor = 2 \] So, \( 2 \times 5 = 10 \).
Step 5: Find the final digit for \( 5^0 \):
\[ 12 - 10 = 2 \] So, \[ d = 2 \]
Step 6: Write the number in base 5:
\[ 137_{10} = 1 \times 5^3 + 0 \times 5^2 + 2 \times 5^1 + 2 \times 5^0 \] So in base 5, this is written as: \[ 1022_5 \]
Key Concept: To convert a decimal number to another base, repeatedly divide by the descending powers of that base and use the quotients as digits, starting from the highest (leftmost) position.
