CALTRX

Long Division Calculator

Divide any two numbers with the complete long division process shown step by step. See the complete solution with step-by-step working and formula explanations.

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Enter your values above to see the results.

Tips & Notes

  • ✓Always verify: quotient × divisor + remainder = dividend. Takes 5 seconds and catches errors.
  • ✓If dividend digit < divisor, write 0 in quotient and bring down next digit together.
  • ✓Remainder must always be less than divisor. If R≥D, you divided too few times.
  • ✓For decimal division: multiply dividend and divisor by same power of 10 to make divisor integer.
  • ✓Long division of polynomials follows identical steps — same process with algebraic terms.

Common Mistakes

  • ✗Forgetting to bring down the next digit after each subtraction step.
  • ✗Writing remainder ≥ divisor — means the partial quotient is too small, add 1.
  • ✗Skipping zeros in quotient. 603÷3: quotient starts 2, then 0 for tens, then 1 → 201.
  • ✗Not verifying answer. Quotient×divisor+remainder must equal dividend exactly.
  • ✗Decimal division: 6.3÷0.7 → multiply both by 10 → 63÷7=9. Never divide by a decimal directly.

Long Division Calculator Overview

Long division is the systematic algorithm for dividing any two numbers, showing quotient and remainder explicitly at each step. It is the manual foundation of all division computation — the same step-by-step process that computers perform in hardware at billions of times per second. Mastering long division builds deep number sense and directly prepares students for polynomial long division in algebra, where the identical algorithm works on algebraic expressions instead of digits.

The four-step algorithm: Divide → Multiply → Subtract → Bring down. Repeat for each digit of the dividend.

EX: 847 ÷ 6 → How many 6s in 8? → 1. Write 1, multiply 1×6=6, subtract 8−6=2, bring down 4 → 24. 24÷6=4. Write 4, multiply 4×6=24, subtract 24−24=0, bring down 7 → 7. 7÷6=1 R1. Write 1. Result: 141 remainder 1
Verify every answer: Quotient × Divisor + Remainder = Dividend
EX: 141 × 6 + 1 = 846 + 1 = 847 ✓ — if this check fails, there is an arithmetic error in the division
Zeros in the quotient — a critical step many students miss. When the partial dividend is smaller than the divisor, write 0 in the quotient:
EX: 603 ÷ 3 → 6÷3=2, bring down 0: 0÷3=0 (write 0!), bring down 3: 3÷3=1 → quotient = 201 not 21
Converting remainder to decimal — append a decimal point and continue dividing:
EX: 847 ÷ 6 = 141 R1 → continue: 10÷6=1 R4, 40÷6=6 R4, 40÷6=6 R4... → 141.1666... = 141.1̄6̄
Dividing by a decimal — multiply both dividend and divisor by the same power of 10 to make the divisor a whole number:
EX: 6.3 ÷ 0.7 → multiply both by 10 → 63 ÷ 7 = 9. Never divide by a decimal directly.
Long division is the bridge between integer arithmetic and decimal arithmetic. When a division does not come out evenly, expressing the result as a decimal means continuing the process with appended zeros — the sequence of remainders determines the decimal digits. Every rational number (any fraction of integers) either terminates or repeats: 1/4 = 0.25 (terminates after 2 digits), 1/3 = 0.333... (repeats immediately), 1/7 = 0.142857142857... (repeats with a 6-digit cycle). Long division also reveals whether a number is divisible by smaller primes — each step shows the remainder, which tells you whether the divisor goes in cleanly. Beyond basic arithmetic, the same algorithm structure underlies polynomial long division in algebra, where polynomials are divided term by term in exactly the same way. The pattern is identical: divide the leading term, multiply back, subtract, bring down the next term, and repeat.

Frequently Asked Questions

Long division follows four repeating steps: Divide, Multiply, Subtract, Bring down — abbreviated DMSB. Example: 847 ÷ 7. Step 1: How many 7s in 8? One. Step 2: 1×7 = 7. Step 3: 8−7 = 1. Step 4: Bring down 4 → 14. Repeat: 14÷7 = 2, 2×7 = 14, 14−14 = 0. Bring down 7 → 7. 7÷7 = 1, 1×7 = 7, 7−7 = 0. Result: 121 remainder 0. Each cycle of DMSB produces one digit of the quotient.

When the divisor does not divide evenly, continue past the decimal point by appending zeros to the remainder. Example: 13 ÷ 4 = 3 remainder 1. Append zero: 10 ÷ 4 = 2 remainder 2. Append zero: 20 ÷ 4 = 5 remainder 0. Result: 3.25 exactly. For repeating decimals like 1 ÷ 3, the remainder cycles: 10 ÷ 3 = 3 r1, 10 ÷ 3 = 3 r1 forever → 0.333... The bar notation 0.3̄ indicates the repeating digit.

The remainder is what is left over after fitting as many whole divisor-groups as possible into the dividend. 25 ÷ 7 = 3 remainder 4 because 7 fits into 25 three times (7×3 = 21), leaving 25 − 21 = 4. The remainder is always less than the divisor — if it equals or exceeds the divisor, another full division step is possible. Remainders appear in programming (modulo operator), scheduling problems, and any situation requiring whole-number division with leftovers.

Polynomial long division follows identical steps but uses terms instead of digits. Divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by that result, subtract from the dividend, and bring down the next term. Example: (x² + 5x + 6) ÷ (x + 2). First step: x² ÷ x = x. Multiply: x(x+2) = x² + 2x. Subtract: (x²+5x+6) − (x²+2x) = 3x+6. Next: 3x ÷ x = 3. Multiply: 3(x+2) = 3x+6. Subtract: 0. Result: x+3.

Estimate by rounding the divisor to the nearest tens or hundreds. 847 ÷ 23: round 23 to 20. 847 ÷ 20 ≈ 42. Try 42 × 23 = 966 (too large). Try 36 × 23 = 828. 847 − 828 = 19 (less than 23, valid remainder). So 847 ÷ 23 = 36 remainder 19. This trial-and-error approach is how the quotient digit is found at each step — experienced calculators recognize that the estimate within 1 or 2 of the correct digit is sufficient.

Long division reinforces place value understanding, estimation, and the relationship between multiplication and division — skills directly used in algebra, fraction simplification, and polynomial arithmetic. The steps of dividing, multiplying, and subtracting mirror how algebraic polynomial division works. Students who master numerical long division find polynomial long division in algebra significantly more approachable because the underlying algorithm is identical.