Long division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.
In Spanish
división larga
Also called
long division method, division algorithm (school method)
Long division handles one digit at a time: divide, multiply, subtract, bring down, and repeat until no digits are left.
Formal definition and notation
Long division computes a ÷ b for a divisorb > 0 by processing the dividend a from its highest place value down. At each place, the current partial dividend p gives one quotient digit d = ⌊ p ÷ b ⌋ and a new partial remainder p − d × b, which is combined with the next digit. The process ends with a = b × q + r, 0 ≤ r < b, or continues past the decimal point to give a decimal quotient.
Key facts
Write each quotient digit directly above the last dividend digit you used, so place value stays lined up.
If the divisor does not fit, write 0 in the quotient before bringing down the next digit: 624 ÷ 6 = 104.
Every subtraction result must be smaller than the divisor; if not, the quotient digit is too small.
Check: divisor × quotient + remainder = dividend.
To get a decimal answer, put a decimal point after the dividend, add zeros and keep going: 7 ÷ 4 = 1.75.
The area model and partial quotients show the same work with bigger chunks; long division records it digit by digit.
The four repeating steps
The four repeating steps
Divide → Multiply → Subtract → Bring down
Divide
how many times does the divisor fit into the current number? Write that digit on top.
Multiply
quotient digit times divisor; write it under the current number.
Subtract
the result must be less than the divisor.
Bring down
the next digit of the dividend, then repeat.
Some students remember the order with “Does McDonald’s Sell Burgers?” When there are no digits left, what remains is the remainder.
Long Division examples, step by step
Example 1: One-digit divisor
Warm-up
Divide 756 ÷ 6.
6 into 7 is 1; 7 − 6 = 1. Bring down 5 to make 15.
6 into 15 is 2; 15 − 12 = 3. Bring down 6 to make 36.
6 into 36 is 6; 36 − 36 = 0.
Quotient 126, remainder 0. Check: 6 × 126 = 756.
Answer:756 ÷ 6 = 126
Example 2: A zero in the quotient
Core
Divide 2,438 ÷ 8.
8 does not go into 2, so use 24: 8 into 24 is 3; 24 − 24 = 0. Bring down 3.
8 into 3 is 0. Write 0 on top; 3 − 0 = 3. Bring down 8 to make 38.
Bring down 2 to make 72. 24 × 3 = 72; 72 − 72 = 0.
Quotient 63, remainder 0. Check: 24 × 63 = 1,512.
Answer: 63 boxes arrive.
Common mistakes
✗ Mistake618 ÷ 3 = 26, skipping the 0 when 3 does not go into 1.
✓ InsteadWrite 0 in the quotient: 618 ÷ 3 = 206.
Each digit brought down needs a quotient digit above it, even when it is 0. Skipping it shifts place value.
✗ MistakeAfter subtracting, getting 9 with a divisor of 7 and bringing down the next digit.
✓ InsteadIncrease the quotient digit by 1, since another 7 fits into 9.
The difference after each subtraction must be less than the divisor.
✗ MistakeWriting the first quotient digit above the first dividend digit in 156 ÷ 4, giving 309.
✓ Instead4 does not go into 1; start above the 5: 156 ÷ 4 = 39.
The first quotient digit sits above the last digit of the first number the divisor fits into.
Not to be confused with
Long Divisionvs.Short division
Short division does the same steps but writes only the small remainders beside the digits, so it suits one-digit divisors. Long division writes every product and subtraction.
Synthetic division is an algebra shortcut for dividing a polynomial by x − c. Long division of numbers works digit by digit.
Quick check: 3 questions
Question 1 of 3
What is 945 ÷ 7?
Show answer and explanation
Answer: A. 7 into 9 is 1 (rem 2); 24 gives 3 (rem 3); 35 gives 5. So 135, and 7 × 135 = 945. 153 swaps digits, 125 slips a step, and 1,035 adds a place.
Question 2 of 3
In long division of 5,236 ÷ 5, what is the remainder?
Show answer and explanation
Answer: D.5 × 1,047 = 5,235, and 5,236 − 5,235 = 1. 6 is bigger than the divisor, and 0 or 2 come from dropping a step.
Question 3 of 3
Which quotient needs a 0 digit in the middle?
Show answer and explanation
Answer: C.1,224 ÷ 4 = 306: after 12 the 2 is too small for 4, so the tens digit is 0. The others give 212, 312 and 115.
Long Division: frequently asked questions
What are the steps of long division?
Repeat four steps. Divide: how many times does the divisor fit into the current number? Multiply that digit by the divisor. Subtract to find what is left. Bring down the next digit. Keep going until there are no digits left; the final difference is the remainder. Check by computing divisor times quotient plus remainder, which should equal the dividend.
How do you do long division with decimals?
If the divisor is a decimal, move its decimal point right until it is a whole number, and move the dividend’s point the same number of places: 4.5 ÷ 0.15 becomes 450 ÷ 15 = 30. Then divide as usual, placing the quotient’s decimal point directly above the dividend’s. Add zeros after the point to keep dividing.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.NBT.B.6, CCSS 5.NBT.B.6, CCSS 6.NS.B.2
Cite this entry
Effortless Math Team. “Long Division.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/long-division/
Flashcard
TermLong Divisiondivide, multiply, subtract, bring down
DefinitionA written method that divides one digit at a time. 872 ÷ 4: 8 gives 2, 7 gives 1, 32 gives 8, so 218.
DividendThe dividend is the number being divided in a division problem. In 56 ÷ 8 = 7, the dividend is 56, the divisor is 8 and the quotient is 7. It is the total you start with before splitting it into equal groups. In a fraction such as 568, the dividend is the numerator.Operations · Grade 3, Grade 4, Grade 5QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Decimal PointA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as “and”.Number System · Grade 4, Grade 5, Grade 6SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3Area ModelAn area model is a rectangle used to multiply or divide, with each side broken into place-value parts. To find 23 × 14, split the sides into 20 + 3 and 10 + 4, find the area of each smaller box (200, 30, 80, 12) and add these partial products: 322.Operations · Grade 3, Grade 4, Grade 5DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5Synthetic DivisionSynthetic division is a shortcut for dividing a polynomial by a linear factor of the form x − k. You write only the coefficients, bring down the first one, then repeat two moves: multiply by k and add. The bottom row gives the quotient’s coefficients, and its last number is the remainder, which equals the polynomial’s value at k.Algebra · Algebra 2, PrecalculusPolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5EstimateAn estimate is an answer that is close to the exact one, and to estimate is to find an answer that is close enough, usually by rounding or using friendly numbers. For example, 49 × 21 is about 50 × 20 = 1000. Estimation helps you check whether an exact answer is reasonable.Problem Solving · Grade 3, Grade 4, Grade 5DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6Standard AlgorithmThe standard algorithm is the usual written column method for adding, subtracting, multiplying or dividing multi-digit numbers. You line up digits by place value, work one place at a time, and regroup (carry or borrow) when a place overflows or runs short. For 478 + 365 it gives 843 in three short steps.Operations · Grade 4, Grade 5, Grade 6