Divisor, dividend, quotient and remainder: what each word means in a division (in 17 ÷ 5 = 3 R 2 the divisor is 5), where each one sits in the long-division layout, the check that catches every mistake (divisor × quotient + remainder = dividend), the second meaning of divisor (a factor), how to find all the divisors of a number and how many there are, the divisibility rules from 2 to 12, and a printable sheet with practice

"Divisor" is searched thirty thousand times a month in the United States, mostly by children in grades 3 to 5 doing long division who have four new words to keep straight, and by older students meeting the second meaning of the word in factors, primes and the greatest common divisor. This page gives the four words with a picture of where each sits in the layout, the one check that catches every long-division mistake, both meanings of divisor, the method for finding every divisor of a number and the formula for how many there are, and the divisibility rules from 2 to 12. The printable has the vocabulary, the layout, the rules and practice on two pages.

Download the division vocabulary and divisibility rules sheet (PDF) (the four words on the long-division layout, the check, the divisibility rules 2 to 12, how to find and count divisors, and twelve practice questions with answers; two pages, free).

The four words

A division has four parts:

In words: dividend ÷ divisor = quotient, remainder r. The words come from Latin: dividend is "the thing to be divided", divisor "the divider", quotient "how many times". A memory for which is which: the dividend is the one that gets divided and ends up in pieces; the divisor is the one doing the dividing.

The same four words apply when the division is written as a fraction: in 17/5, the 17 on top is the dividend (numerator) and the 5 below is the divisor (denominator), and the quotient is 3.4 or 3⅖.

Where each word sits in long division

In the long-division "house" the divisor stands outside on the left, the dividend sits inside under the bar, the quotient is written on top of the bar, and the remainder is what is left at the bottom when the bringing-down is finished:

        quotient  →      3
divisor →  5 ) 17   ← dividend
             -15
            -----
               2   ← remainder

For 452 ÷ 4: the quotient 113 goes on top, the divisor 4 stays outside, the dividend 452 is inside, the remainder is 0. For 1,000 ÷ 7: quotient 142 on top, remainder 6 at the bottom, because 7 × 142 = 994 and 1,000 − 994 = 6.

The check that catches every mistake: divisor × quotient + remainder must equal the dividend. For 1,000 ÷ 7: 7 × 142 + 6 = 994 + 6 = 1,000. If the check fails the division is wrong; if the remainder is 7 or more, the quotient should have been one bigger.

What the remainder means in a word problem

The remainder has to be interpreted, and the right answer depends on the question:

• 50 children, buses hold 12: 50 ÷ 12 = 4 R 2. Round up: 5 buses, because the 2 leftover children need one too.
• 50 cookies, 12 in a box: 4 full boxes and 2 cookies left over.
• 50 dollars shared by 12 people: 4 dollars each and 2 dollars left, or $4.17 each if the remainder is divided too (50 ÷ 12 = 4.1666…).
• Is 50 divisible by 12? No, because the remainder is not 0.

The second meaning: a divisor is a factor

In number theory a divisor of a number is any whole number that divides it with remainder 0. The divisors of 12 are 1, 2, 3, 4, 6 and 12. "Factor" means exactly the same; a proper divisor is any divisor except the number itself; a prime is a number whose only divisors are 1 and itself.

To find all the divisors of a number, test from 1 upwards and write each divisor with its partner (the number divided by it), stopping when the partners meet or cross at the square root:

To count the divisors without listing them, factorise into primes, add 1 to each exponent and multiply. 36 = 2² × 3², so (2 + 1)(2 + 1) = 9 divisors. 60 = 2² × 3 × 5, so (2 + 1)(1 + 1)(1 + 1) = 12. 1,000 = 2³ × 5³, so 4 × 4 = 16 divisors. Only square numbers have an odd number of divisors, because only they have a partner that pairs with itself.

The greatest common divisor (GCD, also called the highest common factor) of two numbers is the largest number that divides both: the divisors of 36 and 60 above share 1, 2, 3, 4, 6 and 12, so the GCD is 12. It is the number that simplifies the fraction 36/60 to 3/5 in one step.

The divisibility rules

A number is divisible by:

These are the rules that make factorising and simplifying fast: a child who knows that 4,521 divides by 3 does not reach for the calculator.

Where it sits in school

The four words arrive with long division: grade 3 and 4 in the United States (Common Core 3.OA and 4.NBT.6, division with remainders), Year 3 and 4 in England, Class 3 and 4 in CBSE, Year 4 in Australia. Divisors as factors, prime factorisation and the greatest common divisor are grade 5 and 6 (6.NS.4), Year 5 and 6, and Class 5 and 6 (Playing with Numbers). The divisibility rules are taught alongside and tested in every entrance and selective exam, because they make mental arithmetic possible.

Practice: twelve questions

1. In 45 ÷ 6 = 7 R 3, name the dividend, divisor, quotient and remainder.
2. 93 ÷ 8: quotient and remainder? Check it.
3. 1,000 ÷ 9: quotient and remainder?
4. A division has divisor 7, quotient 12 and remainder 5. What is the dividend?
5. 125 children, 8 to a table. How many tables?
6. List the divisors of 48.
7. How many divisors has 100? (Use the prime factorisation.)
8. Is 5,238 divisible by 3? By 4? By 6?
9. Is 9,185 divisible by 5? By 9? By 11?
10. What is the greatest common divisor of 24 and 60?
11. Which numbers from 1 to 20 have exactly two divisors?
12. What is the smallest number with exactly six divisors?

Answers: 1) Dividend 45, divisor 6, quotient 7, remainder 3. 2) 11 R 5; 8 × 11 + 5 = 93. 3) 111 R 1. 4) 7 × 12 + 5 = 89. 5) 125 ÷ 8 = 15 R 5, so 16 tables. 6) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. 7) 100 = 2² × 5², so 3 × 3 = 9. 8) By 3 yes (digits add to 18); by 4 no (38); by 6 no. 9) By 5 yes; by 9 no (digits add to 23); by 11 yes (9 − 1 + 8 − 5 = 11). 10) 12. 11) The primes: 2, 3, 5, 7, 11, 13, 17, 19. 12) 12 (divisors 1, 2, 3, 4, 6, 12).

Download the division vocabulary and divisibility rules sheet (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics, for grades 3 to 7 in the US curriculum and the equivalent years in England, Australia and CBSE: long division worked line by line on the shared screen with the four words labelled, the check after every question, then factors, primes and the divisibility rules until they are automatic. The recording and notes after every class, the same tutor every week, US$10 to $13 an hour. The grade 6 math page has the grades; the free 30-minute first class is a real lesson, usually on the homework the child brings.

Questions parents ask

See how we teach this, 1-on-1 online →