Convert 137 to base 5

To convert a number from base 10 to another base, we use the method of successive division. This involves dividing the number by the target base (in this case, 5) repeatedly and recording the remainders until the quotient becomes zero.

Step-by-Step Solution:

1. Divide 137 by 5: $137 \div 5 = 27$ remainder 2

2. Divide 27 by 5: $27 \div 5 = 5$ remainder 2

3. Divide 5 by 5: $5 \div 5 = 1$ remainder 0

4. Divide 1 by 5: $1 \div 5 = 0$ remainder 1

Reading the Result:

To get the final answer, we read the remainders from the bottom to the top (from the last remainder to the first).

• Last remainder: 1
• Third remainder: 0
• Second remainder: 2
• First remainder: 2

Putting these together, we get $1022_5$.

Verification:

We can check our work by converting $1022_5$ back to base 10: $(1 \times 5^3) + (0 \times 5^2) + (2 \times 5^1) + (2 \times 5^0)$ $= (1 \times 125) + (0 \times 25) + (2 \times 5) + (2 \times 1)$ $= 125 + 0 + 10 + 2$ $= 137_{10}$

The calculation is correct.

Exam Tip: When converting to a new base, always remember to write your remainders clearly to the side. If you ever get a remainder larger than the base you are converting to (e.g., getting a 6 when dividing by 5), you know you've made a calculation error!