Theorem

Dividend $=$ divisor $\times$ quotient $+$ remainder

Arithmetic • Number System • Divisibility

Formula

a = b q + r, 0 \leq r < b

For any integer

a

and positive integer

b

there are unique integers

q

and

r

with

a = b q + r

and

0 \leq r < b

. The condition

0 \leq r < b

is what makes

q

and

r

unique, and it is the test a claimed remainder must pass.

How to use it

1Given any three of $a$ , $b$ , $q$ , $r$ , the identity yields the fourth.
2Use $0 \leq r < b$ to reject a misreading: a remainder as large as the divisor means a quotient was understated.
3 $r = 0$ is the statement that $b$ divides $a$ exactly.
4When a question gives a relation between the quotient and the remainder, treat it as a second equation alongside the identity.

A number leaves quotient

14

and remainder

5

on division by

9

. What is the number?

Step-by-step

1
$a = 9 \times 14 + 5$
Why: Apply the identity with $b = 9$ , $q = 14$ , $r = 5$ .
2
$r = 5 < 9$ , so the remainder is legitimate
Why: Check the condition before reporting the answer.
3
$a = 126 + 5 = 131$

Answer

131

Watch out

A remainder is never negative and never reaches the divisor. For a negative dividend the quotient is the one that keeps

r

[0, b)

, so

- 17 = 5 (- 4) + 3

, not

5 (- 3) - 2

Also known as

division algorithm, euclid division lemma, dividend divisor quotient remainder, remainder identity