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Long Division Calculator

dividend divided by divisor equals quotient

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Long Division

Long division breaks a large division problem into a series of smaller steps: divide, multiply, subtract, bring down. You repeat this cycle for each digit of the dividend until all digits have been processed. The calculator supports both remainder and decimal output.

dividend ÷ divisor = quotient

How It Works

Long division breaks a large division problem into a series of smaller steps: divide, multiply, subtract, bring down. You repeat this cycle for each digit of the dividend until all digits have been processed. In remainder mode, you stop once every digit has been brought down and report what is left over. In decimal mode, you keep going by adding zeros after the decimal point so the quotient continues past the remainder.

Example Problem

Divide 845 by 6:

  1. Look at the first digit: 8 ÷ 6 = 1, so write 1 in the quotient and subtract 6 to leave 2.
  2. Bring down the next digit 4 to make 24. Then 24 ÷ 6 = 4, so write 4 and subtract 24 to leave 0.
  3. Bring down the final digit 5. Since 5 ÷ 6 = 0, write 0 in the quotient and leave remainder 5.
  4. The whole-number result is 140 R 5.
  5. If you switch to decimal mode, keep dividing by adding a decimal point and bringing down zeros after the 5.
  6. Continuing gives 845 ÷ 6 = 140.8333…

Result: 140 R 5 (or 140.8333... as a decimal).

Key Concepts

Long division is a step-by-step algorithm for dividing large numbers: divide, multiply, subtract, bring down, and repeat. Each cycle processes one digit of the dividend, building the quotient from left to right. The algorithm can produce either a whole-number quotient with remainder or a decimal result by continuing past the decimal point with appended zeros.

Applications

  • Education: teaching the foundational division algorithm before introducing calculators
  • Mental math: breaking complex division into manageable single-digit steps
  • Programming: implementing arbitrary-precision integer division in big-number libraries
  • Polynomial division: the same algorithmic structure applies to dividing polynomials in algebra
  • Everyday arithmetic: splitting money, quantities, or objects into equal groups while tracking leftovers

Common Mistakes

  • Forgetting to bring down the next digit — skipping a digit produces a quotient that is off by a factor of 10
  • Not placing zeros in the quotient when a brought-down digit is smaller than the divisor — omitting placeholder zeros shifts all subsequent digits
  • Confusing remainder and decimal results — 7 / 4 is either '1 R 3' or '1.75', not both simultaneously

Frequently Asked Questions

How to do long division step by step?

Divide the leftmost digit(s) of the dividend by the divisor, write the quotient digit above, multiply back, subtract, then bring down the next digit. Repeat until no digits remain. The leftover is the remainder.

What is the difference between remainder and decimal division?

Remainder division stops when all digits are processed and reports the leftover (e.g., 7 R 2). Decimal division continues by adding zeros after the decimal point until the result terminates or you reach the desired precision.

Can you divide by a number larger than the dividend?

Yes. The quotient will be 0 with the entire dividend as the remainder (remainder mode), or a decimal less than 1 (decimal mode). For example, 3 ÷ 7 = 0 R 3 or approximately 0.4286.

Why do you sometimes put a 0 in the quotient during long division?

If the current brought-down number is smaller than the divisor, the divisor fits 0 times at that place value. Writing the 0 keeps the quotient digits aligned correctly and prevents later digits from shifting left.

When should I use remainder mode instead of decimal mode?

Use remainder mode when the context cares about whole groups and leftovers, such as splitting 29 students into groups of 4 or counting how many full boxes you can fill. Use decimal mode when you need a precise numeric quotient, such as measurement or finance.

Does long division work with decimals?

Yes. You first shift the decimal point so the divisor becomes a whole number, then continue the usual divide, multiply, subtract, bring-down pattern. This calculator's decimal mode handles the quotient side of that process for you.

What should I do if the decimal never ends?

That means the quotient is a repeating decimal. You can stop at a chosen precision, round the answer, or write the repeating block with a bar notation if your class uses it.

Reference: Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. Pearson.

Long Division Identity

Long division is the step-by-step algorithm behind the division identity shown below. The quotient and remainder always satisfy this relationship exactly, and the decimal form continues the pattern past the decimal point.

Dividend = Divisor × Quotient + Remainder

Where:

  • Dividend — the number being divided
  • Divisor — the number you divide by
  • Quotient — the main answer to the division problem
  • Remainder — what is left over after the last subtraction step

The calculator shows both forms: the whole-number quotient with remainder (e.g. 12 R 1) and the decimal continuation (e.g. 12.0833…).

Worked Examples

Homework Remainder

How do you divide 845 by 6 with a remainder?

A student needs the whole-number quotient and remainder for 845 ÷ 6. The long-division grid shows each divide, multiply, subtract, and bring-down step.

  • 8 ÷ 6 = 1, so write 1 and subtract 6 to leave 2.
  • Bring down 4 to make 24. Then 24 ÷ 6 = 4, subtract 24, remainder 0.
  • Bring down 5. Since 5 ÷ 6 = 0, the quotient digit is 0 and the remainder is 5.
  • Result: 140 R 5, or 140.8333… as a decimal.

Exact Division

What is 125 ÷ 5?

When the dividend is a multiple of the divisor, the remainder is 0. The grid terminates with a clean 0 at the bottom.

  • 5 doesn't fit into 1, so look at 12. 5 ÷ 12 = 2 with remainder 2.
  • Bring down 5 to make 25. 5 ÷ 25 = 5 with remainder 0.
  • Result: 25, exactly.

Divisor Larger Than Dividend

How do you handle 7 ÷ 16?

When the divisor is larger than the dividend, the integer quotient is 0 and the full dividend becomes the remainder.

  • 16 does not fit into 7, so the quotient is 0.
  • The remainder is the original dividend: 7.
  • In decimal form, 7 ÷ 16 = 0.4375.

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