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INTEGER DIVISIBILITY
Tests
Also see the
GCD/LCM Discussion File
The teacher of modern
elementary school mathematics will have to quickly respond to a student's
question as to whether one number the student gives you is divisible by another
number. This skill is brought up
every time you have to factor a set of numbers. It helps to know the following facts.
Definition Divides :
For two integers a and b≠0,
we say that "b divides a" and
write
whenever b is
a factor (divisor with a zero remainder) of a.
When b is a factor of a, we say that a is a multipe of b. We also say that "b does not divide a" and write 
if b is a
divisor of a with a nonzero
remainder.
The following algorithms can
all be explained using the arithmetic properties of the integers, mainly the
properties of our decimal place value system and The Fundamental Rule of
Arithmetic (FTA: Every positive integer is the unique
product of powers of a finite number of primes; uniqueness means independent of
the order of the prime factors). A
complete write up with proofs of the Properties of the Divides relation is
given in the GCD/LCM discussion.
Note: We know
that any nonzero number divides zero and that no number is divisible by zero,
so it is taken for granted that all integers referenced below are nonzero. We start with the rule for the number
2.
Integer Rule
2. A number is divisible by 2 if and only if the number
represented by the right‑hand digit Is divisible by 2.
Examples: l2; 150; 2478; 14.316 are all divisible by
2.
Explanation: 2478=247*10+8. Since 2 is a factor of 10(=2*5), any multiple of 10 is divisible by 2, i.e. 247*10 is divisible by 2. Therefore, any number greater than ten is divisible by 2 if the units' digit Is divisible by 2.
Property a|(b+c) & a|b
a|c
3. A number is divisible by 3 if and only if the sum of
its digits is divisible by 3.
Examples: 12 Is divisible by 3 since the sum of the
digits (1+2=3) is divisible by 3.
Also, 2403 is divisible by 3 since the sum of the digits (2+4+0+ 3=9) is
divisible by 3.
Explanation: Because 1000=999+1, 100=99+1, 10=9+1,
etc., any power of 10 when divided by 3 yields 1 as remainder.
246=2*100+4*10+1=2*(99+1)+4*(9+1)+6=99+9+2+4+6. Since
99+9=3*(33+3) and 12=(2+4+6) are multiples of 3, It follows that
246=(99+9)+(12) is a multiple of 3.
7641=(7*999)+(7*1)+(6*99)+(6*1)+(4*9)+(4*1)+1=(7*999+6*99+4*9)+(7+6+4+1)=3*(7*333+6*33+4*3)+7+6+4+1. Since7+6+4+1=18 is divisible by 3, the
number 7641 is also divisible by 3. Note: If 999 Is evenly divisible by 3, then
7*999 is also evenly divisible by 3, since a factor of a number is also a
factor of any of its multiples.
4. A number is divisible by 4 if and only if the number
represented by the two right‑hand digits is divisible by 4.
Examples: 24; 7432; 159,52 are divisible by 4.
Explanation: 3428=3400+2A. Since 4 is a factor of
100=4*25, any multiple of 100 is divisible by 4. Hence, when determining
whether a number is divisible by 4, we need only be concerned with whether the
number represented by the two right‑hand digits is divisible by 4.
5. A number is divisible by 5 if and only if it ends in
5 or 0.
Examples: 75; 130; 2435; 7945; are divisible by 5.
Explanation: 2435=2430+5.
Since
5 Is a factor of 10=2*5, any multiple of 10 is divisible by 5. Hence, when
determining whether a number is divisible by 5, we need only be concerned with
whether the units' digit is divisible by 5.
6. A number is divisible 6 if and only if it is
divisible by 2 and by 3.
6=2*3, and 2 and 3 are primes and have only 1 as a
common divisor, gcd(2,3)=1
Examples: 42; 144; 2316; 17,994; are divisible by 6.
7.
Double and subtract. That is, double the right-hand digit and subtract it from
the number defined by the remaining digits when the unit's digit is removed.
Repeat this double and subtract process until it becomes evident that 7 does or
does not divide the result. This
may be best seen with an example.
We want to check if 7 divides 18053.
Question: Does 7 divide 18053 evenly?
Double
the value of the most right hand digit that is 3 and you get 6.
Subtract 6 from the number 1805 defined when unit's digit 3 is removed
fm 18053. 1805-6=1799
Repeat: double 9 to get
18 and subtract 18 from 179 (1799 w/ unit's digit 9 removed) 179-18=161
Repeat: double one to get two and subtract from
16 16-2=14
We
know that 7|14 so we know that 7|1805
The
question is answerable at any point where is it evident that 7 does or does not
divide the difference.
If
we saw that 7 divides 161, then we could have stopped then and concluded that 7
does divide 18053.
Does
7 divide 6524. Double 4 to get 8
and subtract from 94 is not divisible by 7 because 9-(2*4)=1 is not divisible
by 7. However, 7 divides 6524
because 652–8=644.
Continuing we see that 64-8=56 and we know that 7 divides 56. So we know that 7 divides 6524.
11. An
integer is divisible by 11 if and only if it satisfies the even minus odd
test. That is, an integer is
divisible by 11 if and only if the sum of the digits in the columns that are
even powers of 10 minus the sum of the digits in the columns that are
odd powers of 10 is divisible by11 (or vice versa, take odd-even because if a|b
then a|-b).
So 11 divides an integer if and only if 11|[(sum digits in even columns minus
sum digits in odd columns], abbreviated, 11|(sum even – sum odd).
8. A
number is divisible by 8 If and only if the number represented by the three
right‑hand digits is divisible by 8.
Examples: l000, 4456 and 928 are all divisible by 8.
Explanation:
73,960=73,000+960. Since 8=
is a factor of 1000=
, any multiple of 1000 is divisible by 8. Hence, when
determining whether a number is divisible by 8, we need only be concerned with
whether the number represented by the three right‑hand digits is
divisible by 8. Similarly, 2
or 16 would be determined by the number of the four
right-hand digits, but it would be smarter to use a calculator if you had one
for checking numbers of greater than three digits.
9. A number is divisible by 9 if and only if the sum of
Its digits is divisible by 9.
Examples:
18,279; 4563; 19,467; are divisible by 9.
Explanation:
Any power of ten when divided by 9 gives 1 as remainder. The explanation Is
similar to that for the rule of divisibility for 3.
10. A number is divisible by 10 if and only If it ends
in 0.
Examples:
60; 150; 2250; 94,370; are divisible by 10.
Evaluation:
The student should try to explain the rule of divisibility for 10.
11. An integer is divisible by 11 (7 or 13) if and only
if 11|[(sum even minus
sum odd] . See discussion
given in the Rule for 7 for a more complete explanation.
12. A number is divisible by 12 if and only if it is
divisible by 3 and by 4.
Examples:
1788;
4044; 35,220; are divisible by 12.
Explanation: 12
= 3 x 4, and 3 and 4 are relatively prime, i.e. gcd(3,4)=1.
General: m | n An integer n is divisible by an integer m if and only if n is
divisible by all of the relatively prime factors of m.
For
example: 6= 2*3 and gcd(2,3)=1, so 6| m iff both 2| m and 3| m.
Condensed Summary of the Divisibility
tests
a. An integer is divisible by 2, 5, or 10
if and only if its units digit is divisible by 2, 5, or 10, respectively.
b. An integer is divisible by 4 if and
only if the last 2 digits of the integer represent a number divisible by 4.
c. An integer is divisible by 8 if and
only if the last 3 digits of the integer represent a number divisible by 8.
d. An integer is divisible by 3 or by 9 if
and only if the sum of its digits is divisible by 3 or 9, respectively.
e.
An integer is divisible by 7, 11or 13 if and only if the sum of the digits in
the odd place value columns minus the sum of the digits in the even place value
columns is divisible by 11.
Symbolically, 7,11 or
13|(sum odd place value digits) – (sum even place value digits). Note: a|b ®a|(-b), so odd-even or even-odd does not matter.
Alternate
rule for divisibility by 7. Seven
divides an integer if and only it 7 divides the result after you double and
subtract.
f.
m | n An integer n is divisible by an integer m if and only if n is
divisible by all of the relatively prime factors of m.
For example, 6 divides an integer if and only if the integer is divisible by both of the relatively prime factors 2 and 3
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