Long Division Calculator

Long division is the pencil-and-paper algorithm for dividing one number (the dividend) by another (the divisor) when mental math or a basic calculator is not enough. The process estimates how many times the divisor fits into leading digits of the dividend, subtracts, brings down the next digit, and repeats until no digits remain. The final answer has two parts: the quotient (how many whole times the divisor fits) and the remainder (what is left over, smaller than the divisor).

Example: 162 ÷ 7. Seven fits into 16 two times (2 × 7 = 14). Subtract to get remainder 2, bring down 2 to make 22. Seven fits into 22 three times (3 × 7 = 21). Subtract to get remainder 1. Quotient = 23, remainder = 1, written as 23 R1. You can also express the result as a mixed number 23 1/7 or a decimal 23.142857… by continuing the algorithm past the decimal point. This calculator shows the classic stacked format so students can compare their handwritten work line by line.

Another classic case: 100 ÷ 3. Three fits into 10 three times with remainder 1. Bring down 0 (append a zero after exhausting integer digits) to get 10 again — the remainder cycle repeats, producing the repeating decimal 33.3̄. Quotient = 33, remainder = 1. Integer division truncates toward zero; floor division would differ for negative inputs. Understanding remainder versus decimal form matters in modular arithmetic, clock arithmetic, and cryptography where only remainders are stored.

The division algorithm rests on the identity dividend = divisor × quotient + remainder, with 0 ≤ remainder < |divisor|. Verify any result by multiplying back: 7 × 23 + 1 = 162. For divisors that do not evenly divide, decimal expansion continues by appending zeros to the remainder and repeating the subtract-multiply cycle. Terminating decimals occur only when the divisor's prime factors (after simplifying with the dividend) are limited to 2 and 5; otherwise the decimal repeats or eventually cycles.

Connect long division to the fraction calculator when expressing remainders as proper fractions, the GCF calculator when simplifying divisors before dividing, and the basic calculator for quick decimal checks. In algebra, polynomial long division mirrors this digit-by-digit structure. In number theory, the remainder is the result of the modulo operation used in hash functions and checksums.

Enter dividend and divisor as integers or decimals. The tool scales fractional inputs to integers internally for exact quotient and remainder, then displays a decimal approximation when the division is not exact. Use paper-format output to learn the algorithm, remainder form for discrete problems, and decimal form for measurements. Division by zero is undefined and is rejected with a clear message.

Practice problems often mix formats: word problems asking how many 7-inch ribbons fit into 162 inches yield the same computation as symbolic 162 ÷ 7. Checking your work by multiplying quotient and divisor, then adding remainder, catches transcription errors before they propagate through multi-step homework. When remainders must be expressed as fractions, divide the remainder by the divisor and simplify — here 1/7 is already in lowest terms, which you can confirm with the GCF calculator.