Long Division Calculator

The Long Division Calculator shows how a dividend is divided by a divisor one place at a time. It reports the quotient and, for integer division, any remainder while exposing the repeating divide–multiply–subtract–bring-down cycle.

Zeros inside the quotient matter

At each place, determine how many times the divisor fits into the current partial dividend. Write that quotient digit, multiply it by the divisor, subtract, and bring down the next digit. A remainder must be smaller in magnitude than the divisor.

Dividend = divisor × quotient + remainder, with a remainder smaller in magnitude than the divisor under the usual whole-number convention.

Enter the values in their mathematical roles

Enter the dividend first and a nonzero divisor second. Choose whether the answer should stop with a remainder or continue into decimal places. When decimals are present, shift both numbers by the same power of ten to make the divisor whole.

The governing relationship

Dividend = divisor × quotient + remainder, with a remainder smaller in magnitude than the divisor under the usual whole-number convention.

A compact example

157 ÷ 12 gives quotient 13 and remainder 1 because 12×13+1=157.

Valid inputs and boundary cases

Division by zero is undefined. Negative signs determine the quotient sign, while remainder conventions can vary across programming and mathematics contexts. Specify the required convention for negative integer division.

Read the answer in context

The divisor cannot be zero, and decimal continuation may repeat.

An independent check

Multiply the quotient by the divisor and add the remainder. The result should reproduce the dividend. For decimal answers, expect a small difference if the quotient was rounded.

Errors that change the answer

• Reversing dividend and divisor
• Placing a quotient digit over the wrong column
• Forgetting to bring down a zero
• Allowing a remainder at least as large as the divisor
• Treating a rounded decimal as exact

Continue with a related calculator

Continue with the Rounding Calculator or Quadratic Formula Calculator. For everyday arithmetic and order-of-operations checks, use the Basic Calculator.

What this numerical result cannot decide

The calculator demonstrates numeric long division. Polynomial long division uses a related layout but operates on terms and powers rather than decimal digits, so it requires a different tool and validation method.

A second way to understand the method

When the divisor does not fit into a place after the quotient has started, write a zero in that quotient position before bringing down the next digit. Omitting it changes place value. For example, 1,005÷5=201, not 21.

From remainder form to decimal form

For 23÷4, integer division gives 5 remainder 3. To continue, place a decimal point in the quotient, append a zero to the remainder, and divide 30 by 4. This produces 7 remainder 2; another zero produces 5, so the decimal result is 5.75.

Why a repeating decimal appears

In 1÷6, the first decimal digit is 1 with remainder 4, then 40÷6 gives 6 with remainder 4 again. Because the same remainder returns, the digit 6 repeats forever. Recording remainders makes the repeating pattern clearer than copying a long display of identical digits.

Place-value alignment is the core of the method

Each quotient digit belongs above the corresponding dividend place. Multiplying and subtracting underneath that same place preserves value. Graph paper or ruled columns can help learners keep digits aligned, especially when zeros occur inside the dividend. A misplaced quotient digit changes the answer by a factor of ten even if every local multiplication is correct.

Long Division Calculator FAQ

What is the remainder in 17÷5?

The quotient is 3 with remainder 2 because 5×3+2=17.

Can long division produce repeating decimals?

Yes. A repeated remainder generates a repeating digit cycle.

Why move both decimal points?

Multiplying dividend and divisor by the same power of ten preserves their ratio.

For additional worked mathematics, see the free OpenStax Prealgebra 2e reference.