Long Division Calculator
Divide any two integers with step-by-step long division. Shows quotient, remainder, decimal result, and the full long division process as text.
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How to use this calculator
Long division finds how many times the divisor fits into the dividend (quotient) and what is left over (remainder). The decimal form continues dividing the remainder.
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Enter the dividend (the number being divided) in the first field.
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Enter the divisor (the number you are dividing by) in the second field.
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The quotient, remainder, decimal result, and step-by-step long division are shown.
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Negative inputs are handled — the sign of the quotient follows integer division rules.
Frequently asked questions
What is the difference between quotient and remainder?
The quotient is how many whole times the divisor fits into the dividend. The remainder is what is left after those whole divisions. For 847 ÷ 13: 13 fits 65 times (quotient), with 847 − 65×13 = 847 − 845 = 2 left over (remainder 2). Check: 13 × 65 + 2 = 847. ✓
How does long division work?
Long division processes the dividend one digit at a time from left to right. Bring down digits, divide by the divisor for each partial quotient digit, subtract the product, and repeat with the remainder. The process generates each digit of the quotient in sequence and is the algorithm taught in elementary school for multi-digit division.
What happens when the remainder is 0?
When the remainder is 0, the divisor divides the dividend exactly — the divisor is a factor of the dividend. The decimal result is a whole number or terminating decimal. For 840 ÷ 4 = 210 with remainder 0; 4 is a factor of 840.
How do I interpret the decimal result from long division?
The decimal extends the division beyond the integer quotient. After the remainder, you append a decimal point and continue: bring down zeros and keep dividing. This produces a decimal that may terminate (if the divisor has only factors 2 and 5) or repeat (if it has other prime factors). For 847÷13: decimal ≈ 65.153846…, which repeats because 13 has prime factor 13 (neither 2 nor 5).
Long division calculator — quotient, remainder, and step-by-step
The long division algorithm
Long division is a systematic method for dividing large numbers. The dividend's digits are processed left to right: at each step, find the largest multiple of the divisor that fits into the current partial dividend, write it as the next quotient digit, subtract, and bring down the next digit. Repeat until all digits are processed. The method requires only multiplication and subtraction knowledge, making it accessible once those operations are mastered.
Division theorem and remainder
The Division Theorem (Euclidean Division) states that for any integers a (dividend) and b ≠ 0 (divisor), there exist unique integers q (quotient) and r (remainder) with 0 ≤ r < |b| such that a = b×q + r. The verification row in this calculator checks this identity. The remainder is always non-negative and strictly less than the divisor — which is why 10 ÷ 3 gives remainder 1 (not 4 or 7), and why the next division step would need a larger partial dividend.
Long division and polynomial division
The same long division algorithm extends to polynomials: polynomial long division divides P(x) by D(x) to find Q(x) and R(x) such that P(x) = D(x)×Q(x) + R(x), where degree(R) < degree(D). This is used in algebra to simplify rational expressions, find asymptotes of rational functions, and in the Factor Theorem (if R = 0, then D(x) is a factor of P(x)). The structural parallel between integer and polynomial division is deep and pedagogically important.
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