The Division Calculator computes the quotient, remainder, and full decimal with long division steps.
Shows step-by-step long division, results as simplified fractions and mixed numbers, and divisibility rule checks (2, 3, 4, 5, 6, 8, 9, 10).
Calculator information
📋 How to use this calculator
- Enter the number to be divided (dividend) in the first field.
- Enter the divisor, which must not be zero.
- Choose how to display the result: integer with remainder, decimal, or mixed fraction.
- Enable the 'long division' option to see each step of the standard long-division algorithm.
- Verify results using divisibility rules (e.g., a number is divisible by 3 if the sum of its digits is divisible by 3).
- Tip: turn on the repeating-decimal display to see patterns like 1/3 = 0.333...
🧮 Euclidean Division Algorithm
dividend = divisor x quotient + remainder, 0 <= remainder < |divisor|
- dividend = the number being divided
- divisor = the number you divide by (not equal to 0)
- quotient = the integer result of the division
- remainder = the leftover after division
- Decimal = remainder / divisor (long-division process until the pattern repeats or terminates)
If the dividend is negative, the Euclidean modulo definition keeps the remainder non-negative.
💡 Worked example: Long division of 437 / 12
Given:- Dividend = 437
- Divisor = 12
Steps:- Take the first two digits, 43: 43 / 12 = 3 remainder 7 (3 x 12 = 36).
- Bring down the next digit, 7, to form 77.
- 77 / 12 = 6 remainder 5 (6 x 12 = 72).
- Integer quotient = 36, remainder = 5.
- Continuing as decimal: 50 / 12 = 4 remainder 2; 20 / 12 = 1 remainder 8; the pattern continues. 437/12 = 36.4166...
Result: Integer result 36 remainder 5; mixed number 36 5/12; decimal 36.41666... (repeating 6).
❓ Frequently asked questions
What is long division?
Long division is the standard step-by-step algorithm for dividing multi-digit numbers by hand. The dividend goes inside the division bracket, the divisor outside, and you work digit by digit from left to right. It is typically introduced in 3rd and 4th grade in U.S. elementary math curricula such as Common Core.
How can you quickly tell whether a number is divisible by 3?
A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 4,521 has digits summing to 4+5+2+1 = 12, and 12 is divisible by 3, so 4,521 is divisible by 3. A similar rule applies for 9 (the digit sum must be divisible by 9).
What is the difference between quotient, remainder, and modulus?
The quotient is the integer result of a division, the remainder is what is left over, and 'modulus' in programming is usually synonymous with remainder. The '%' operator in many programming languages takes the sign of the dividend, while the strict mathematical modulo is always non-negative.
Why is division by zero undefined?
The equation a = 0 x q has no unique solution for a not equal to 0, and infinitely many solutions for a = 0. Because there is no consistent value of q, division by zero is left undefined. In analysis, the limit of 1/x as x approaches 0 tends to plus or minus infinity.
What is a repeating decimal and how do you write it?
A repeating decimal is the result of a division whose tail repeats a fixed pattern, such as 1/7 = 0.142857142857... The formal notation places a bar above the repeating digits. Any fraction whose denominator has a prime factor other than 2 or 5 produces a repeating decimal.
📚 Sources & references
Last updated: May 11, 2026