check arithmetical work via the method of CASTING
OUT NINES.
Divisibility by 10
Any number N can be written in the form N = 10a + b,
where b is the final digit of N. Thus a number is divisible 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 + b is divisible
by 17 only if 10a + b – 51b = 10(a – 5b) is. This, in
turn, shows that N is 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 multiple 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 example. 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 SUBTRACTION. 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 a is divided by a number b to produce
a quotient q, a ÷ b = q, then a is called the dividend
and b the divisor. The quotient can also be expressed as
a FRACTION, a/b, or a RATIO, a:b. In general, the quotient q of two numbers a and b satisfies 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. Unfortunately, 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, FRACTIONs, REAL
NUMBERS, and COMPLEX NUMBERS. In all settings, the
number 1 acts as an identity element—provided it
operates as a divisor: a ÷ 1 = a for 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; EUCLIDEAN ALGORITHM; FACTOR; FACTORIZATION; FACTOR THEOREM; RATIONAL FUNCTION; REMAINDER THEOREM.
divisor Another name for FACTOR.
146 division
OUT NINES.
Divisibility by 10
Any number N can be written in the form N = 10a + b,
where b is the final digit of N. Thus a number is divisible 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 + b is divisible
by 17 only if 10a + b – 51b = 10(a – 5b) is. This, in
turn, shows that N is 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 multiple 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 example. 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 SUBTRACTION. 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 a is divided by a number b to produce
a quotient q, a ÷ b = q, then a is called the dividend
and b the divisor. The quotient can also be expressed as
a FRACTION, a/b, or a RATIO, a:b. In general, the quotient q of two numbers a and b satisfies 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. Unfortunately, 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, FRACTIONs, REAL
NUMBERS, and COMPLEX NUMBERS. In all settings, the
number 1 acts as an identity element—provided it
operates as a divisor: a ÷ 1 = a for 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; EUCLIDEAN ALGORITHM; FACTOR; FACTORIZATION; FACTOR THEOREM; RATIONAL FUNCTION; REMAINDER THEOREM.
divisor Another name for FACTOR.
146 division