check arithmetical work via the method of
CASTING
OUT NINES
.
Divisibility by 10
Any number Ncan be written in the form N= 10a+ b,
where bis the final digit of N. Thus a number is divisi-
ble by 10 only if its final digit is a multiple of 10. We
have:
A number is divisible by 10 only if its final
digit is a zero.
Divisibility by 11
The numbers 100, 1000,… alternately leave remainders
of 1 and –1 when divided by 11. (For example, 100 is 1
more than a multiple of 11, but 1,000 is 1 less.) Thus
the remainder of a number when divided by 11 is
obtained as the alternate sum of its digits. For example,
69,782, which equals 6 ×10,000 + 9 ×1,000 + 7 ×100
+ 8 ×10 + 2 ×1, leaves a remainder 6 ×1 + 9 ×(–1) +
7 ×1 + 8 ×(–1) + 2 ×1 = 6 – 9 + 7 – 8 + 2 = –2 when
divided by 11. (This is equivalent to a remainder of 9.)
We have:
The remainder of any number when divided by
11 is the alternate sum of its digits. Thus a
number is divisible by 11 only if the alternate
sum of its digits is a multiple of 11.
Divisibility by 12
A number is divisible by 12 only if it is divisible by
both 3 and 4. Thus we have:
A number is divisible by 12 only if its final two
digits represent a two-digit multiple of 4, and
the sum of all the digits of the number is a
multiple of 3.
The divisibility rule for 7 can be extended to other
numbers as well. For example, N= 10a+ bis divisible
by 17 only if 10a+ b– 51b= 10(a– 5b) is. This, in
turn, shows that Nis divisible by 17 precisely when
quantity a– 5b, obtained by deleting and subtracting 5
times the final digit, is divisible by 17. Notice here that
51 is the first multiple of 17 that is 1 more than a mul-
tiple of 10.
In the same way we can use that fact that 111 is
the first multiple of 37 that is 1 more than a multiple of
10 to obtain a similar divisibility rule for 37, for exam-
ple. Divisibility rules for all
PRIME
numbers, except 2
and 5, can be created this way.
division The process of finding the
QUOTIENT
of two
numbers is called division. In elementary arithmetic,
the process of division can be viewed as repeated
SUB
-
TRACTION
. For instance, 60 divided by 12 equals 5
because 12 can be subtracted from this number five
times before reaching zero: 60 – 12 – 12 – 12 – 12 – 12
= 0. We write: 60 ÷ 12 = 5. Division can also be
described as the process of finding how many subsets
or magnitudes are contained within a set or given
quantity. For instance, 5 ÷ 1/2 = 10 because 10 lengths
of one-half are contained in a length of 5 units.
If a number ais divided by a number bto produce
a quotient q, a ÷b= q, then ais called the dividend
and bthe divisor. The quotient can also be expressed as
a
FRACTION
, a/b, or a
RATIO
, a:b. In general, the quo-
tient qof two numbers aand bsatisfies the equation
q×b= a. Thus division may also be thought of as
the inverse operation to
MULTIPLICATION
. Thus, since
5×12 = 60, for instance, 5 is indeed the quotient of 60
and 12. This reasoning also shows that, since 0 ×b= 0
for any nonzero number b, we have 0 ÷ b= 0. Unfortu-
nately, one cannot give meaning to the quantity 0 ÷ 0.
(Given that 53 ×0 = 0, we may be forced to conclude
that 0 ÷ 0 = 53. At the same time, since 117 ×0 = 0,
we also have that 0 ÷ 0 = 117. We have inconsistency.)
It is also not possible to give meaning to the term a÷0
for any nonzero value a. (If a÷0 = q, then q×0 = a,
yielding a
CONTRADICTION
.)
The
LONG DIVISION
algorithm provides a means to
divide large integers. The process of division can be
extended to
NEGATIVE NUMBERS
,
FRACTION
s,
REAL
NUMBERS
, and
COMPLEX NUMBERS
. In all settings, the
number 1 acts as an identity element—provided it
operates as a divisor: a÷ 1 = afor all numbers a.
The symbol ÷ is called the “obelus” and first
appeared in print in Johann Heinrich Rahn’s 1659 text
Teutsche algebra.
See also
DIVISIBILITY RULES
;
DIVISOR
;
DIVISOR OF
ZERO
; E
UCLIDEAN ALGORITHM
;
FACTOR
;
FACTORIZA
-
TION
;
FACTOR THEOREM
;
RATIONAL FUNCTION
;
REMAIN
-
DER THEOREM
.
divisor Another name for
FACTOR
.
146 division