Long Division Calculator

Dividend:

Enter up to 48 digits; spaces, commas, and underscores are accepted as grouping marks.

Divisor:

Use a positive whole number; the remainder will always be smaller than this value.

Decimal continuation:

Choose 0 for quotient-and-remainder form, or continue through at most 24 decimal digits.

Digit grouping:

Keep separators for easier scanning or show ungrouped digits for worksheet matching.

Step	Stage	Working number	Quotient digit	Product	Remainder	Explanation	Copy
{{ row.step }}	{{ row.stage }}	{{ row.working }}	{{ row.digit }}	{{ row.product }}	{{ row.remainder }}	{{ row.note }}

Check	Value	Meaning	Copy
{{ row.label }}	{{ row.value }}	{{ row.note }}

Long division turns a large division problem into a sequence of small place-value decisions. At each step, the divisor is fitted into the current working number, the matching product is subtracted, and the remainder is carried into the next digit. The written work matters because it shows where every quotient digit came from.

Three numbers define whole-number division. The dividend is being divided, the divisor is the positive size of each group, and the quotient counts complete groups. Anything left is the remainder, which must be at least zero and smaller than the divisor.

Ways to express a division result
Form	What it preserves	Typical use
Quotient and remainder	Complete groups and the exact leftover	Whole objects, sharing, and arithmetic checks
Decimal quotient	Place-value continuation after the whole-number step	Measurement and approximate comparison
Reduced fraction	The exact ratio in simplest terms	Exact arithmetic when a decimal repeats or is truncated

A nonzero remainder can be continued into decimal places by appending a zero and repeating the same divide, multiply, and subtract cycle. Some divisors eventually produce a remainder of zero, giving a terminating decimal. Others return to a remainder seen earlier; from that point, the decimal digits repeat in the same cycle.

Stopping after a chosen number of decimal digits creates a truncated value, not a rounded one. For exact work, keep the quotient-and-remainder identity or the reduced fraction. A decimal display is useful for scale, but a short continuation can hide a repeating cycle or omit later digits.

How to Use This Tool:

Enter non-negative whole numbers and decide how far, if at all, the remainder should continue into decimals.

Enter the Dividend and a positive Divisor. Spaces, commas, and underscores may group digits, but signs and decimal points are not accepted.
Choose Decimal continuation from 0 to 24 places. Use 0 when quotient-and-remainder form is the intended answer.
Read the quotient and remainder, then follow Division steps to see each product and subtraction. Use Answer check to confirm the recomposition and exact reduced fraction.

Interpreting Results:

The whole-number quotient and remainder are exact. The recomposition should reconstruct the dividend, and the remainder must satisfy 0 ≤ remainder < divisor.

Read the decimal status before copying a decimal quotient. Terminating means the remainder reached zero. Repeating cycle means a remainder returned. Truncated means the selected limit was reached before either condition was proved.

Technical Details:

Whole-number long division implements Euclidean division in base 10. Each quotient digit is the largest digit whose product with the divisor does not exceed the current working number. Subtraction produces the remainder carried to the next place.

Formula Core:

For dividend N, positive divisor D, quotient Q, and remainder R, exact whole-number division satisfies:

\begin{array}{lcl} N & = & D \times Q + R \\ 0 & \leq & R < D \end{array}

Transformation Core:

Decimal continuation transforms the current remainder into the next working number by multiplying it by 10. Integer division supplies the next digit, and subtraction supplies the next remainder.

\begin{array}{lcl} W_{i} & = & 10 R_{i - 1} \\ R_{i} & = & W_{i} - D q_{i} \end{array}

Each nonzero remainder is recorded before the next decimal digit is produced. If a later step returns to the same remainder, all following states repeat, so the intervening digits form the cycle. If the remainder becomes zero, the decimal terminates. The selected continuation limit stops the process after at most 24 digits and does not round the last digit.

Long division input and calculation rules
Rule	Exact behavior
Integer size	Dividend and divisor are each limited to 48 normalized digits.
Allowed values	The dividend may be zero. The divisor must be greater than zero.
Arithmetic	Whole-number operations use exact integer arithmetic rather than floating-point division.
Reduced fraction	Dividend and divisor are divided by their greatest common divisor.
Digit grouping	Thousands separators change display only; they do not change arithmetic or exported values.

For 9,876 divided by 37, the exact whole-number result is 266 remainder 34 because 37 × 266 + 34 = 9,876. Continuing six places gives 266.918918. The remainder returns after three decimal digits, so 918 is the detected repeating cycle rather than a rounded ending.