Divide with long division — see every subtract-and-bring-down step, the quotient, and the remainder laid out. Free and step-by-step.
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Long division divides multi-digit numbers by repeated divide–multiply–subtract–bring down cycles, producing the quotient digit by digit and leaving a remainder. It is the most intricate of the four basic algorithms — and the most satisfying when the steps click. Our free Long Division Calculator performs the full tableau: enter dividend and divisor, and get the quotient, remainder, and every intermediate subtraction.
The algorithm is repeated estimation: at each step, guess how many times the divisor fits into the current working number, multiply back, subtract, and bring down the next digit. Each quotient digit is verified by the subtraction — if the remainder exceeds the divisor, the guess was too small; if it goes negative, too big.
Long division also reveals decimal structure: continuing past the decimal point (bringing down zeros) generates the decimal expansion, exposing terminating vs repeating behavior. The quotient–remainder form and the decimal form are two views of one division.
Long division computes q and r with a = bq + r (0 ≤ r < b) by processing the dividend one digit at a time. Maintain a running remainder: bring down the next digit (×10 + digit), find the largest qd with qd·b ≤ working number, subtract, repeat. Each step’s qd is one quotient digit; the final running remainder is r.
The estimation step — “how many times does b go into the working number” — is the skill long division teaches. Guess via leading digits (e.g., 13 into 123: 13×9 = 117 ≤ 123, 13×10 = 130 > 123, so 9). The subtraction verifies: remainder must be less than b, else the guess was too small; negative means too big.
The check a = bq + r is the division algorithm itself — every division must satisfy it. The calculator displays the check explicitly, so each result is self-verifying. If the check fails, a step went wrong.
Continuing past the units (bringing down zeros after a decimal point) extends the quotient into decimal places. The remainders then cycle or terminate, revealing the decimal expansion — long division is also the engine behind fraction-to-decimal conversion.
For each digit: working = 10·remainder + digit; qd = ⌊working ÷ b⌋; subtract qd·b; new remainder. Result: a = bq + r.
b ≠ 0. Each qd is one quotient digit; the final remainder satisfies 0 ≤ r < b.
Digits: 1→12→123→1234→12345. 13 into 123: 9 (9×13=117), remainder 6. Bring down 4: 64; 13×4 = 52, remainder 12. Bring down 5: 125; 13×9 = 117, remainder 8. Quotient 949, remainder 8. Check: 13×949+8 = 12337+8 = 12345 ✓.
7 into 10: 1, remainder 3. Bring down 0: 30; 7×4 = 28, remainder 2. Quotient 14, remainder 2. Check: 7×14+2 = 100 ✓. As decimal: 14.285714… — the 142857 cycle appears.
25 into 100: 4, remainder 0. Bring down 0: 0; 25×0 = 0. Quotient 40, remainder 0 — exact division. Round divisors often divide cleanly.
2 into 7: 3, remainder 1. Quotient 3, remainder 1. Check: 2×3+1 = 7 ✓. Continuing: bring down 0 → 10, 2×5 = 10 → 3.5.
A warehouse packs 12,345 items into boxes of 13: 12,345 ÷ 13 = 949 remainder 8 — 949 full boxes and 8 left over. The quotient plans the shipment; the remainder flags the partial box.
Splitting a $100 bill seven ways: 100 ÷ 7 = 14 remainder 2 — $14 each with $2 left for the tip jar. The remainder has direct physical meaning.
A programmer distributes 1,000,000 records across 25 shards: 1,000,000 ÷ 25 = 40,000 exactly, remainder 0 — perfectly balanced. The zero remainder confirms no shard is shortchanged.
The division algorithm states: for integers a, b > 0, there exist unique q, r with a = bq + r and 0 ≤ r < b. Long division is the constructive proof — it finds q and r digit by digit. Uniqueness is why the check a = bq + r is decisive: only one (q, r) pair can satisfy it.
The running remainder invariant is the key: after processing k digits, (prefix value) = b·(quotient so far) + remainder, with remainder < b·10. Each step preserves this while extending the quotient by one digit. Induction over digits proves the algorithm — the calculator’s step list is the induction unrolled.
This same structure powers polynomial long division (divide by leading terms, subtract, bring down next term) and the Euclidean algorithm for GCDs. Long division is the first member of a family of “divide, multiply, subtract, bring down” algorithms spanning arithmetic and algebra.
Long division’s complexity is O(n²) for n-digit numbers with the schoolbook method — each of n quotient digits needs an O(n) multiply-subtract. Faster algorithms (Newton–Raphson division, Barrett reduction) reach O(n log n), which is why computers do not long-divide. But the schoolbook method remains the best human algorithm: its steps are checkable, its errors localizable, and its tableau teaches place value with every digit.
The remainder’s sign in computing follows the dividend: −7 ÷ 3 gives quotient −3 remainder 2 in mathematics (remainder non-negative), but many programming languages give −2 remainder −1 (truncated division). The calculator uses the mathematical convention — know which your tools use.
Short division is long division compressed: for single-digit divisors, the subtract-and-bring-down happens mentally, writing only quotient digits and small remainder annotations. 945 ÷ 5 = 189 via short division in seconds. Learn long first (it shows everything), then compress to short for speed.
Divisibility shortcuts bypass the tableau for special divisors: ÷5 = ×2 ÷10 (840 ÷ 5 = 1680 ÷ 10 = 168); ÷25 = ×4 ÷100; ÷125 = ×8 ÷1000. These are fraction equivalences (1/5 = 2/10) applied as division — the calculator’s long division is the general method; these are the fast lanes.
The remainder theorem for polynomials mirrors integer division: dividing P(x) by (x − a) leaves remainder P(a). Synthetic division is polynomial long division compressed — same tableau, same subtract-and-bring-down rhythm, one degree higher.
9 into 9: 1, remainder 0. Bring down 9: 1, remainder 0. Bring down 9: 1. Quotient 111, remainder 0. Repdigits divide into repdigit quotients.
16 into 50: 3 (48), remainder 2. Bring down 0: 20; 16×1 = 16, remainder 4. Quotient 31, remainder 4. Check: 16×31+4 = 496+4 = 500 ✓.
8 into 1: 0, remainder 1. Bring down 0: 10; 8×1 = 8, remainder 2. Bring down 0: 20; 8×2 = 16, remainder 4. Bring down 0: 40; 8×5 = 40, remainder 0. Quotient 0.125 — the decimal emerges digit by digit.
Divide 945 ÷ 15 by long division by hand, writing each subtract-and-bring-down step. Verify quotient, remainder, and the check with the calculator.
Compute 1000 ÷ 8 two ways: by long division, and by halving three times (÷2 ÷2 ÷2). Why do both give 125?
Divide 1 ÷ 7 to six decimal places by continuing past the point. What repetend emerges? Relate it to the fraction-to-decimal converter.
A freight forwarder loads 12,345 kg into containers holding 13,000 kg? No — into pallets of 13 kg capacity each (small parcels): 12,345 ÷ 13 = 949 remainder 8. So 949 full pallets and 8 kg hand-carried. The quotient plans the pallets; the remainder plans the exception.
When the client adds 5 kg (12,350 total): 12,350 ÷ 13 = 950 exactly, remainder 0. The forwarder notices the roundness — 13 × 950 = 12,350 — and double-checks the client’s figure against the check line. Long division’s built-in verification (a = bq + r) is the quality control.
The Long Division Calculator divides non-negative whole numbers with the classic digit-by-digit tableau, reporting the quotient, remainder, the verification a = bq + r, the decimal value, and a per-digit step list showing each working number, quotient digit, multiplication, and subtraction.
Process the dividend one digit at a time: form the working number (10×remainder + next digit), find how many times the divisor fits (qd), multiply back, subtract, and bring down the next digit. Repeat until digits run out.
Divide (estimate qd), Multiply (qd×divisor), Subtract, Bring down. Repeat. The mnemonic “DMSB” — Dad, Mother, Sister, Brother — helps students remember.
The subtraction verifies it: the remainder after subtracting must be non-negative and less than the divisor. Too big → qd was too large; remainder ≥ divisor → qd was too small.
Quotient 949, remainder 8. Check: 13×949 + 8 = 12,345. As a decimal, about 949.615.
Yes — keep bringing down zeros after a decimal point. The quotient extends into decimal places, revealing terminating or repeating structure.
To distinguish it from short division (mental, small divisors). “Long” refers to writing out every subtract-and-bring-down step in the full tableau.
The quotient is 0 with remainder equal to the dividend (e.g., 7 ÷ 13 = 0 remainder 7). The algorithm handles it: no digit of the dividend reaches the divisor.
a ÷ b = a/b. Continuing the division past the decimal point computes the fraction’s decimal expansion — long division is fraction-to-decimal conversion by hand.
Cite this page as: Long Division Calculator. Sigma Calculator. Retrieved October 10, 2026, from https://sigmacalculator.net/math-calculators/long-division-calculator/
Last updated: October 10, 2026