UNIT 2. DIVISIBILITY. INTEGERS. POWERS.

DIVISIBILITY. BASIC CONCEPTS.

The factors of a whole number are the numbers that divide into it exactly.
By the way , factors are also called divisor

For example, the factors of 18 are 1, 2, 3, 6, 9, and 18.

A number is called a prime number if it has exactly two factors (i.e. 1 and itself).
The first few prime numbers are 2, 3, 5, 7, 11, 13, … . Note that 1 is not a prime number.

PRIME NUMBERS TO 100

2,3,5,7,11,13,17,19,23,29,

31,37,41,43, 47,53,59,61,

67,71,73,79,83,89,97


The multiples of a number are all the numbers that it will go into exactly.

For example, the multiples of 6 are 6, 12, 18, 24, 30, … .

A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.

By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!

COMPOSITE NUMBERS UP TO 20

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20

You can write any composite number as a product of primes factors. This is called prime factorization.

FACTOR TREE

Factor trees are used to break a number up into its prime factors.

Example: Draw the factor tree for 60.

We begin by finding two whole numbers that multiply to make 60, for example 6 × 10: We keep splitting up each number in this way until we reach a prime number. Once a prime number is reached, we circle it and then that part of the diagram is finished. The completed factor tree looks as follows:


The circled numbers multiply together to make 60, i.e. 60 = 2 × 2 × 3 × 5. This product is called the prime factorisation of 60.

Divisibility Tests

Divisibility Tests
Example
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
168 is divisible by 2 since the last digit is 8.
A number is divisible by 3 if the sum of the digits is divisible by 3. 168 is divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is divisible by 3.
A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
316 is divisible by 4 since 16 is divisible by 4.
A number is divisible by 5 if the last digit is either 0 or 5. 195 is divisible by 5 since the last digit is 5.
A number is divisible by 6 if it is divisible by 2 AND it is divisible by 3.
168 is divisible by 6 since it is divisible by 2 AND it is divisible by 3.
A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
7,120 is divisible by 8 since 120 is divisible by 8.
A number is divisible by 9 if the sum of the digits is divisible by 9.
549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9.
A number is divisible by 10 if the last digit is 0.
1,470 is divisible by 10 since the the last digit is 0.

Divisibility by 11: Divisibility by 11 is the most interesting of the above tests (7 will be studied below). We do two sums (the odd numbered digits and the even numbered digits), subtract one sum from the other, and see if this is divisible by 11. By the way, if we end up with zero, then that is divisible by 11. We can repeat that process, just as we did with 3.

Let's look at an example:


    34871903
                3+8+1+0=12
                4+7+9+3=23
                                    23-12=11
                                                    Is divisible by 11


We can, of course, do the summing in different orders. In fact we can just go from left to right adding and subtracting alternate digits: 3-4+8-7+1-9+0-3=-11 (divisible by 11).

LCM (Least Common Multiple) and GCF (Greatest Common Factor)

To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.

Find the GCF and LCM of 84 and 140 :

 The factor tree for 84 is  The factor tree for 140 is


My prime factorizations are:
  • So: 84 = 2 × 2 × 3× 7
  • Also, 140 = 2 × 2 × 5 × 7
This orderly listing, with each factor having its own column, will do most of the work for me

The Greatest Common Factor, the GCF(Mcd), is the biggest number that will divide into (is a factor of) both 84 and 140. In other words, it's the number that contains all the factors common to both numbers. In this case, the GCF is the product of all the factors that 84 and 140 have in common.

This orderly listing, with each factor having its own column, will do most of the work for me.

 84 2 2 3  7
 140 2 2  5 7
 GFC 2 2   7

Then the GCF is 2 × 2 × 7 = 28.

On the other hand, the Least Common Multiple, the LCM, it is the smallest number that contains both 84 and 140 as factors, the smallest number that is a multiple of both these values. Then it will be the smallest number that contains one of every factor in these two numbers.

 84 2 2 3  7
 140 2 2  5 7
 LCM 2 2 3 5 7

Then the LCM is 2 × 2 × 3 × 5 × 7 = 420.

By using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly with only one factor for each column (you can have 2s columns, 3s columns, etc, but a 3 would never go in a 2s column), and then carry the needed factors down to the bottom row.

For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry down all the factors, regardless of how many or few values contained that factor in their listings.

INTEGERS DEFINITION
  • The number line goes on forever in both directions. This is indicated by the arroes.
  • Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
  • Whole numbers less than zero are called negative integers. thes numbers are to the left of zero on the number line.
  • The integer zero is neutral. It is neither positive nor negative.
  • The sign of an integer is positive (+) or negative (-), except zero, wich has no sign.
  • Two integers are opposites if they are each the same distance away from zero, but opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labelled as opposites.


ABSOLUTE VALUE OF AN INTEGER

The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a numbere n by writing n in between two vertical bars: | n |

Examples:

| 6 | = 6;  | -12 | = 12;   | 0 | = 0;   | 1234 | = 1234;   | -1234 | = 1234

INTEGERS: OPERATIONS WITH SIGNED NUMBERS

Before you do ANY computation, determine the OPERATION!
Then follow the instructions for THAT operation.

ADDITION

Do the numbers have the SAME SIGN?

YES
NO
Same sings:
Find the SUM
Different signs:
Find the DIFFERENCE
(-3) + (-6) = (-9)
(+4) + (+5) = (+9)
(+5) + 8-7) = (-2)
(-4) + (+6) = (+2)

Either way Keep the sign of the "LARGE number". "LARGER" is used here as a quick (but mathematically imprecise) way to describe the integer with the greater Absolute Value (ie. distance from zero). In each of the examples above, the SECOND integer has a greater Absolute Value.


SUBTRACTION

First, change the SUBTRACTION problem to an ADDITION problem:
First, copy the problem exactly ( -6 ) - ( +2 ) =
1. The first number stays the same ( -6 )
2. change the operation ( -6 ) +
3. Switch the NEXT SIGN ( -6 ) + ( -2 )
4. Follow the rules for addition ( -6 ) + ( -2 ) = ( -8 )

Subtract means: ( +2 ) - ( -6 )
    Add the opposite: ( +2 ) + ( +6 ) = ( +8 )

Subtract means: ( -7 ) - ( -3 )
    Add the opposite: ( -7 ) + ( +3 ) = ( -4 )

Subtract means: ( +4 ) - ( +9 )
    Add the opposite: ( +4 ) + ( -9 ) = ( -5 )

MULTIPLICATION OR DIVISION

First, DO the multiplication or division.
Then determine the sign: count the number of negative signs ... Are there an EVEN number of negative signs?
  • YES (an EVEN number of negative signs) the answer is POSITIVE
  • NO (an ODD number of negative signs) the answer is NEGATIVE
Example:

First, copy the problem exactly: ( -2 ) · ( -4 ) · ( -6 ) =
Do the multiplication or division. 2 · 4 · 6 = 48
Count the number of negative signs... Determine the signs of the answer:
Are there an EVEN number of negatives?
If YES, the answer is POSITIVE, otherwise, the answer is NEGATIVE.
A total of TRHEE NEGATIVES, trhee is NOT EVE (it's odd)
So the answer is NEGATIVE ( -2 ) · ( -4 ) · ( -6 ) = ( -48 )

Examples

(4) : (2) · (6) = 12
    A total of ZERO NEGATIVES
    So the answer is POSITIVE

(4) : (-2) · (6) = -12
    A total of ONE NEGATIVE
    One is NOT EVEN (it's odd)
    So the answer is NEGATIVE

(-4) : (2) · (-6) = 12
    A total of TWO NEGATIVES
    Two is EVEN
    So the answer is POSITIVE

ORDER OF OPERATIONS

Problem: Evaluate this arithmetic expression: 18 + 36 : 32

In the last year, we learned how to evaluate an arithmetic expression with more than one operation according to the followings rules:

Rule 1: Simplify all operations inside parentheses
Rule 2: Perform all multiplications and divisions, working from left to right
Rule 3: Perform all additions and subtractions, working from left to right

However, the problem above includes an exponent, so we cannot solve it without revising our rules.

Rule 1: Simplify all operations inside parentheses
Rule 2: Simplify all exponents, working from left to right.
Rule 3: Perform all multiplications and divisions, working from left to right
Rule 4: Perform all additions and subtractions, working from left to right

We can solve the problem above using our revised order of operations.

Evaluate this arithmetic expresion:
18 + 36 : 32
Simplify all exponents (Rule 2)     18 + 36 : 9
Division (Rule 3)                           18 + 4
Addition (Rule 4)                           22

POWERS AND ROOTS. SQUARES, CUBES AND THEIR ROOTS

Squaring a number 32 means "3 squared‟, or 3 · 3. The small 2 is an index number, or power. It tells us how many times we should multiply 3 (base) by itself. Similarly 72 means "7 squared‟, or 7 · 7. And 102 means "10 squared‟, or 10 · 10.
So, 12 = 1 · 1 = 1; 22 = 2 · 2 = 4 ; 32 = 3 · 3 = 9; 42 = 4 · 4 = 16; 52 = 5 · 5 = 25.
1, 4, 9, 16, 25… are known as square numbers.

When the index number is greater than three we say 'to the power of'.
For example: 37 is 'three to the power of seven',

45 is 'four to the power of five'.

Exponent Properties

1. Product of like bases: To multiply powers with the same base, add the exponents and keep the common base.
 
22 · 23 = 2 · 2 · 2 · 2 · 2 = 25
                                                 xm · xn = xm+n

2. Quotient of like bases:
To divide powers with the same base, subtract the exponents and keep the common base.

Write two examples with numbers and variables.

3. Power to a power: To raise a power to a power, keep the base and multiply the exponents.

Write two examples with numbers and variables.

4. Product to a power: To raise a product to a power, raise each factor to the power.

Write two examples with numbers and variables.

5. Quotient to a power: To raise a quotient to a power, raise the numerator and the denominator to the power.

Write two examples with numbers and variables.

Exponents one and zero


Notice that 31 is the product of only one 3, which is evidently 3.
Also note that 35 = 3·34. Also 34 = 3·33.
Continuing this trend, we should have
31 = 3 · 30.

Another way of saying this is that when n, m, and n − m are positive (and if x is
not equal to zero), one can see by counting the number of occurrences of x that
Extended to the case that n and m are equal, the equation would read

since both the numerator and the denominator are equal. Therefore we take this as the definition of x0.

This leads to the following rule:

Any number to the power 1 is itself.
Any nonzero number to the power 0 is 1; one interpretation of these powers is
as empty products. The case of 00 is discussed be


Negative exponents

A negative exponent means to divide by that number of factors instead of multiplying. So 4-3 is the same as 1/43, and x-3 =1/ x3

As you know, you can't divide by zero, so
x ≠ 0 when x=0, x-n is undefined.

SCIENTIFIC NOTATION

For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers).

The format for writing a number in scientific notation is fairly simple:
First digit of the number followed by the decimal point and then all the rest of the digits of the number, times (10 to an appropriate power). The conversion is fairly simple.

Example: Write 12400 in scientific notation.

This is not a very large number, but it will work nicely for an example. To
convert this to scientific notation, I first write "1.24". This is not the same
number, but (1.24)(10000) = 12400 is, and 10000 = 104. Then, in scientific
notation, 12400 is written as 1.24 · 104.

SQUARE ROOTS


The opposite of a square number is a square root.
We use the symbol
to mean square root.
So we can say that


4 = 2 ( 4 is called radicand) and 25 = 5.

However, this is not the whole story, because (–2) · ( –2 ) is also 4,
and ( –5 ) · ( –5 ) is also 25.

So, in fact, 4 = 2 or –2. And 25 = 5 or – 5.

Remember that every positive number has two square roots.

CUBING A NUMBER


13 = 1 · 1 · 1 = 1;  2 · 2 · 2 means "2 cubed‟, and is written as 23.
33 = 3 · 3 · 3 = 27; 43 = 4 · 4 · 4 = 64; 53 = 5 · 5 · 5 = 125

1, 8, 27, 64, 125… are known as cube numbers.

CUBE ROOTS

The opposite of a cube number is a cube root. We use the symbol 3 to mean
cube root.
So is 3 8 = 2 and3 27 is 3.