A 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.
The remainder is what cannot form another full group. Dividing by 5, it can only be 0, 1, 2, 3 or 4.
Formal definition and notation
For whole numbersa and b > 0, there are unique whole numbers q (the quotient) and r (the remainder) with a = b × q + r and 0 ≤ r < b. This is the division algorithm. The remainder is 0 exactly when b is a factor of a. The remainder can also be written as a fraction of the divisor: 23 ÷ 5 = 435.
Symbol
Read it as
Meaning
4 R 3
four remainder three
quotient 4 with 3 left over
23 mod 5 = 3
23 mod 5 equals 3
the remainder alone, used in computing and number theory
Key facts
Check a division with a remainder by multiplying and adding: 5 × 4 + 3 = 23.
If your remainder is equal to or bigger than the divisor, the quotient is too small: one more group fits.
A remainder can be written three ways: 4 R 3, the mixed number435, or the decimal 4.6.
In word problems you must decide what the remainder means: round the quotient up (extra bus needed), drop the remainder (only full boxes count), or report the leftovers.
Division with remainder
Division with remainder
dividend = divisor × quotient + remainder
remainder
a whole number from 0 up to divisor − 1
Remainder = dividend − divisor × quotient. For 100 ÷ 7: 7 × 14 = 98, so the remainder is 100 − 98 = 2.
Remainder examples, step by step
Example 1: Find the remainder
Warm-up
Find the quotient and remainder of 38 ÷ 6.
The largest multiple of 6 not over 38 is 6 × 6 = 36.
38 − 36 = 2, and 2 < 6.
So 38 ÷ 6 = 6 R 2.
Answer: The quotient is 6 and the remainder is 2: 6 R 2.
Example 2: Three ways to write it
Core
Write 47 ÷ 4 with a remainder, as a mixed number and as a decimal.
4 × 11 = 44, and 47 − 44 = 3: 11 R 3.
The remainder over the divisor is 34: 1134.
34 = 0.75, so 11.75.
Answer:11 R 3, or 1134, or 11.75.
Example 3: What does the remainder mean?
Test-style
A van holds 8 people. How many vans are needed for 70 people, and how many seats will be empty?
70 ÷ 8 = 8 R 6: 8 full vans carry 64 people.
The 6 people left still need a ride, so round up to 9 vans.
Seats: 9 × 8 = 72, so 72 − 70 = 2 are empty.
Answer: 9 vans are needed, with 2 empty seats.
Common mistakes
✗ Mistake29 ÷ 4 = 6 R 5
✓ Instead29 ÷ 4 = 7 R 1, since 4 × 7 = 28.
A remainder of 5 is bigger than the divisor 4, so another group of 4 still fits.
✗ Mistake17 ÷ 5 = 3 R 2 = 3.2
✓ Instead3 R 2 = 325 = 3.4.
The remainder is a count of leftovers, not a decimal digit. Divide it by the divisor to get the decimal part.
✗ Mistake45 students, 10 per table: 4 tables, because 45 ÷ 10 = 4 R 5.
✓ Instead5 tables are needed; the last table seats the 5 extra students.
Leftover people still need a place, so the quotient rounds up.
A difference is the result of subtracting: 23 − 5 = 18. A remainder is what is left after removing as many full groups as possible.
Quick check: 3 questions
Question 1 of 3
What is the remainder of 58 ÷ 9?
Show answer and explanation
Answer: A.9 × 6 = 54 and 58 − 54 = 4. 6 is the quotient, 5 is an off-by-one slip, and 13 is too big to be a remainder with divisor 9.
Question 2 of 3
When a number is divided by 6, which remainder is impossible?
Show answer and explanation
Answer: D. Remainders with divisor 6 run from 0 to 5. A remainder of 6 would form another full group.
Question 3 of 3
Pencils come in boxes of 12. A teacher has 100 pencils. How many full boxes can she fill, and how many pencils are left?
Show answer and explanation
Answer: A.12 × 8 = 96 and 100 − 96 = 4. Only full boxes count, so 8 boxes with 4 left. 9 boxes would need 108 pencils.
Remainder: frequently asked questions
How do you find the remainder?
Divide to get the whole-number quotient, multiply it by the divisor, and subtract that from the dividend. For 75 ÷ 8: the quotient is 9 because 8 × 9 = 72 and 8 × 10 = 80 is too big. Then 75 − 72 = 3, so the remainder is 3. Always check that the remainder is less than the divisor.
How do you turn a remainder into a fraction or decimal?
Write the remainder over the divisor. 19 ÷ 4 = 4 R 3 becomes 434, and since 34 = 0.75, the decimal is 4.75. The remainder 3 is not the decimal part; it is 3 out of the 4 needed for another whole group.
DivisorA 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 5Whole 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 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 5DivisionDivision 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 5NumberA 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 2Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2Mixed NumberA mixed number has a whole number and a fraction together, such as 234, which means 2 + 34. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 234 = 114.Number System · Grade 4, Grade 5, Grade 6Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4DifferenceA 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 3DividendThe 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 5SubtractionSubtraction 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 3Long DivisionLong 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.Operations · Grade 4, Grade 5, Grade 6Divisibility RulesDivisibility rules are shortcuts that tell whether one whole number divides another with no remainder, without doing the division. A number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.Number System · Grade 4, Grade 5, Grade 6Remainder TheoremThe Remainder Theorem says that when a polynomial p(x) is divided by x − a, the remainder equals p(a). So substituting x = a replaces the division. For p(x) = x2 + 1 divided by x − 3, the remainder is p(3) = 10. A remainder of 0 means x − a is a factor.Algebra · Algebra 2, Precalculus