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Natural Numbers and Integers |
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Order of operations,
brackets
as grouping symbols |
The
divisibility of integers |
Prime
and composite numbers |
The
integer prime factorization |
The
greatest common divisor/factor (GCD/GCF) |
The
least common multiple (LCM) |
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Order
of operations -
brackets
as grouping symbols |
When expressions have more than one operation, we have to follow
rules for the order of operations. First do all multiplication and
division operations working from left to right. Next do all addition
and subtraction. |
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Example: |
-
8 -
3 ·
(-
4)
+ 15
¸ (-
5) =
-
8 -
(-
12)
+
(-
3) =
-
8
+ 12
-
3
= 1 |
|
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We often use grouping symbols, like
brackets, to help us
organize complicated expressions into simpler ones. Do operations
in brackets
and other grouping symbols first. If there are grouping
symbols within other grouping symbols do the innermost first. |
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Example: |
-
3
+ 2
· [-
2 -
3 ·
(-
4 ¸
2
+ 1)]
=
-
3 + 2
· [-
2 -
3 ·
(-
2 +
1)] =
-
3
+ 2
· [-
2 -
3 ·
(-
1)]
=
|
|
=
-
3
+ 2
· [-
2 +
3]
=
-
3
+ 2
=
-
1 |
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The
divisibility of integers |
A
divisor of an integer n, also called a
factor of n, is an integer which
evenly divides n
without leaving a remainder. Numbers divisible by 2 are called even and those that are not are called
odd. |
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Example: |
7 is
divisor of 21 because 21 ¸ 7
=
3. We also say 21 is
divisible by 7 or 21 is a multiple of 7. |
|
The
divisors
of 21 are { 1, 3, 7, 21 }. |
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The
divisibility rules |
|
A number is divisible by
2 if the last digit is divisible by 2. |
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|
A number is divisible by
3 if the sum of its digits is divisible by 3. |
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A number is divisible by
4 if the number given by the last two digits is divisible by 4. |
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A number is divisible by
5 if the last digit is 0 or 5. |
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A number is divisible by
6 if it is divisible by 2 and by 3. |
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A number is divisible by
7 if the result of subtracting twice the last digit from the number (with the last digit removed) is divisible by 7 (e.g. 581 is divisible by 7 since
58 -
2 · 1 =
56 is divisible by 7). |
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A number is divisible by
8 if the number given by the last three digits is divisible by 8. |
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A number is divisible by
9 if the sum of its digits is divisible by 9. |
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A number is divisible by
10 if the last digit is 0. |
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Prime
and composite numbers |
A
prime number is a natural number greater than one whose only
positive divisors are one and itself. |
A
natural number that is greater
than one and is not a prime is called a composite number. The
sequence of prime numbers begins 2, 3, 5, 7, 11, 13, 17, 19, 23, ... |
|
The
integer prime-factorization |
Every positive integer greater than 1 can be factored as
a product of prime numbers.
|
Given a number
n divide by the first prime from the list of primes, if it does not divide
cleanly, divide n by the next prime, and so on.
|
|
Example: |
Thus, the prime factors of 350
= 2
· 5
· 5
· 7.
|
|
|
350
| 2 |
175
| 5 |
35
| 5 |
7 | 7 |
1
| |
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The
greatest common divisor (GCD) |
The greatest common divisor of two integers is the largest integer
that divides both numbers. |
Greatest common divisors can be computed by determining the prime
factors of the two numbers and comparing the factors, as in the following example: |
|
Example: |
72
= 2
· 2
· 2
· 3
· 3,
90 = 2
· 3
· 3
· 5,
=>
GCD(72, 90) = 2
· 3
· 3
= 18. |
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Two numbers are called
relatively prime if their greatest common divisor equals 1.
Thus, for example |
12
= 2
· 2
· 3,
25 = 5
· 5
=>
GCD(12, 25) = 1. |
|
The
least common multiple (LCM) |
A
common multiple is a number that is a multiple of two or more
numbers. |
The least common multiple of two integers a and
b
is the smallest positive integer that is a multiple of both
a and
b. To
find the least common multiple of two numbers: - first list the prime
factors of each number, then multiply each factor the greatest
number of times it occurs in either number. If the same factor occurs more than once in both numbers, multiply the factor the greatest
number of times it occurs. |
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Example: |
Let's find the
LCM of 24, 54 and 60, |
|
the prime factors: 24 = 2
· 2
· 2
· 3,
54 = 2
· 3
· 3
· 3,
60 = 2
· 2
· 3
· 5, |
==> LCM(24, 54, 60) = 2
· 2
· 2
· 3
· 3
· 3
· 5 = 1080. |
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Beginning
Algebra Contents A |
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