division with remainder
In the divides function from the last
section, we employed a while
loop. The eventual termination of the loop
depends on the following principle.
Archimedes' principle. For an integer \( a \) and a nonzero integer \( b \), there exists a multiple of \( b \) which is greater than or equal to \( a \).
This principle is equivalent to the division algorithm, which we will prove.
proposition 2 (division algorithm). For an integer \( a \) and a nonzero integer \( b \), there are unique integers \( q \) and \( r \) such that \( a = bq + r \) and \( 0 \le r < b. \)
Terminology:
 \( a \) is called the dividend.
 \( b \) is called the divisor.
 \( q \) is called the quotient.
 \( r \) is called the remainder.
Before proving the division algorithm, here are a few examples.
 Dividing \( 71 \) by \( 20 \) gives \( 71 = 20(3) + 11 \)
 Dividing \( 71 \) by \( 20 \) gives \( 71 = 20(3) + 11 \)
 Dividing \( 71 \) by \( 20 \) gives \( 71 = 20(4) + 9 \)
 Dividing \( 71 \) by \( 20 \) gives \( 71 = 20(4) + 9 \)
proof of division algorithm: If \( a \ge 0 \), consider the set \( S \), consisting of integers of the form \( a  bn \) where \( n \) is an integer, ie, \[ S = { a  bn: n \in \mathbf{Z} }. \] The intersection of \( S \) the with nonnegative integers is nonempty, since \( a \) itself is nonnegative. By the wellordering principle, there must be a unique smallest nonnegative element of \( S \), say \( r = a  bq \). Since \( r  b \) is a smaller element of \( S \), it must be negative, hence \( r < b \).
If \( a < 0 \), then \( a \) is positive, which was proven above to be expressable as \( a = bQ + R. \) If \( R = 0 \), set \( q = Q \) and \( r = 0. \) Otherwise, set \( r = R + b \) and \( q = Q  b / b. \) In either case, we have \[ a = bQ  R = bq + r \] with \( 0 \le r < b. \)
in code
In most programming languages, the implementation of a division and remainder (modulus) operator likely will not match the specification given by the division algorithm: when the dividend or divisor is negative, it may result in a negative remainder.
In ruby, for example, integer division is floor division, so the quotient
a / b
is computed by rounding down the rational number \( a / b \).
Therefore, the sign of the remainder a % b
is the same as the sign of the
divisor b
.
a  b  a / b  a % b 

71  20  3  11 
71  20  4  9 
71  20  4  9 
71  20  3  11 
In rust, on the other hand, integer division is truncation, so the quotient
a / b
is computed by rounding the rational number \( a / b \) toward zero.
Therefore, the sign of the remainder a % b
is the same as the sign of the
dividend a
.
a  b  a / b  a % b 

71  20  3  11 
71  20  3  11 
71  20  3  11 
71  20  3  11 
exercise: Write a
div_rem
function that takes a dividenda
and a nonzero divisorb
as inputs, and outputs a quotientq
and a remainderr
satisifying the division algorithm.
The division algorithm itself gives the specifications to validate that
div_rem
is implemented correctly. Following is a test in ruby that asserts
the properties from the division algorithm for some test inputs. Note that I
am specifically suppressing test inputs where the arguments could be zero since
div_rem(a, b)
is undefined when b == 0
.
def test_div_rem
test_inputs(:include_zero => false).each do (a, b)
quo, rem = div_rem(a, b)
assert 0 <= rem
assert rem < b.abs
assert_equal a, b*quo + rem
end
end
Click the expand button to see an implementation of div_rem
in ruby.
divisibility and remainders
The division algorithm is an extension of the idea of divisibility, as this proposition makes clear.
proposition 3. A nonzero integer \( b \) divides an integer \( a \) if and only if the division of \( a \) by \( b \) yields a remainder of \( 0 \).
proof: If the division algorithm gives \( a = bq + r \) with \( r = 0\), then \( a = bq \), which implies the \( b \mid a \).
Conversely, if \( b \mid a \), then \( a = bn \) for some \( n \). Since the division algorithm produces a unique quotient and remainder, it must be true that \( q = n \) and \( r = 0 \).
exercises
Compute by hand the quotient and remainder when dividing:

\( 345 \) by \( 14 \)

\( 872 \) by \( 77 \)

\( 7373 \) by \( 205 \)

\( 762 \) by \( 11 \)