By Talmadge Walker
Updated Aug 30 2022

Computations in a base other than ten can feel daunting, especially when the familiar multiplication tables of base‑ten no longer apply. Long division still relies on estimation, multiplication, and subtraction, but the process requires a fresh approach. This guide walks you through the method, using a base‑seven example to illustrate each step.

Step 1: List the Divisor’s Multiples in the New Base

Begin by creating a table of single‑digit multiples of the divisor, expressed in the target base. For instance, dividing 1431 (base 7) by 23 (base 7) starts with:

• 23 × 1 = 23
• 23 × 2 = 46
• 23 × 3 = 102
• 23 × 4 = 125
• 23 × 5 = 151
• 23 × 6 = 204

Because we’re working in base 7, we only need multiples up to 6. Listing these multiples ahead of time eliminates the need to recall multiplication facts in a non‑decimal system.

Step 2: Select the Highest Multiple Not Exceeding the Leading Digits

From the list, choose the largest multiple that does not surpass the initial digits of the dividend. In our example, 125 (base 7) fits, while 151 and 204 are too large. Write the corresponding digit, 4, above the dividend, since 23 × 4 = 125.

Step 3: Subtract the Chosen Multiple

Subtract 125 (base 7) from the leading portion of the dividend (143 in base 7), yielding 15 (base 7).

Step 4: Bring Down the Next Digit

Lower the following digit of the dividend—here, the 1—to form the new temporary remainder of 151 (base 7).

Step 5: Repeat Until the Remainder is Smaller Than the Divisor

Consult the multiples table again. 23 × 5 = 151, so place the digit 5 above the dividend next to the 4, then subtract 151 from 151 to leave a remainder of zero.

Step 6: Record Any Non‑Zero Remainder

If the final remainder were greater than zero, you would write it as Rremainder to the right of the quotient. In this case, the remainder is zero, so the final answer is 45 (base 7).

TL;DR (Too Long; Didn’t Read)

When performing long division in a non‑decimal base, remember that standard base‑ten multiplication facts don’t apply. Verify your result by converting the divisor, dividend, and quotient back to base ten. Calculators typically don’t handle custom bases unless specifically designed to do so. For bases above ten, use alphabetic symbols for digits 10, 11, 12, etc.