Long Division Problems

Long division is a standard method for dividing large numbers that breaks the problem into a series of smaller, manageable steps. You set up the problem using the division bracket, then follow the repeating cycle of Divide, Multiply, Subtract, and Bring Down (DMSB) until all digits of the dividend have been processed. Long division applies to whole numbers, decimals, and even polynomials. Students typically learn long division in grades 3–4 and continue using it through high school when they encounter polynomial long division.

When the dividend contains a decimal, write it in the quotient directly above its position in the dividend. When the divisor is a decimal, first shift the decimal point in both the divisor and dividend the same number of places to the right until the divisor becomes a whole number, then divide normally. This adjustment does not change the result because you are scaling both numbers by the exact same power of 10. Practice this technique with our division worksheets for grades 5–6.

Place a 0 in the quotient for that position and bring down the next digit. This is one of the most common wrong results learners produce—they skip the zero, which shifts all subsequent digits and gives an incorrect answer. For example, in the long division problem 612 ÷ 6, the right answer is 102, not 12. To avoid this mistake, always check whether the divisor fits into each group of digits before moving on.

Polynomial long division handles any polynomial divisor and follows the same DMSB process as numerical long division, but with algebraic terms instead of single digits. Synthetic division is a streamlined shortcut that only applies when the divisor is a linear binomial of the form (x − c). It uses only the coefficients, which makes calculations faster, but the approach is less general. Both are covered in our long division worksheets for grades 9–12.

Long division builds number sense, estimation skills, and a deep understanding of how division actually works—skills that are essential from grade school through calculus. Long division is also the foundation for polynomial division, which most calculators cannot perform. Understanding the algorithm helps with mental math, checking calculations, and related topics like the Remainder Theorem, Factor Theorem, and partial fraction decomposition.

When you divide by a 2-digit divisor, the key is estimation. Look at the first two or three digits of the dividend and determine how many times the divisor fits. For example, to divide 4,752 by 36, start by estimating how many times 36 goes into 47. Write your estimate above the dividend, then complete each divide-multiply-subtract-bring-down cycle. Repeat until you have processed every digit. If your estimate is wrong, adjust up or down and try again. Practice long division on paper until it feels automatic.

We offer free long division worksheets for grades 3 through 12. You can generate division worksheets filtered by grade level, the range of digit divisors, and whether you want remainders or exact results. Each worksheet comes with a complete answer key. Practicing long division is one of the best ways to build accuracy—grab a worksheet and a pencil to get started.

The most common long division mistakes include: forgetting to place a zero in the quotient when the divisor does not go into the next number, making errors during the subtract step, and bringing down the wrong digit. Some learners also skip the final step, which gives an incorrect remainder. To push past these errors, solve each long division problem carefully, double-check every calculation, and verify that your result is right.