# Number Theory Lesson Notes

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```					                                        Number Theory

Even Numbers
All even numbers are divisible by 2. This means that eve numbers can be divided into two
equal-size groups or divided into pairs, with no leftovers. Every even number has 0, 2, 4, 6, and
8 in its ones place.

Odd Numbers

Odd numbers are not divisible by 2. When an odd number is divided into two equal-size groups
or divided into pairs, there is always 1 left over. Every odd number has 1, 3, 5, 7, or 9 in its ones
place.

Even and Odd Number Patterns

Examples                         Computation                           Result

72 + 38 = 110              even number = even number                  even number
71 + 38 = 109              odd number + even number                   odd number
71 + 37 = 108               odd number + odd number                   even number
72 + 37 = 109              even number + odd number                   odd number

72 – 38 = 34              even number – even number                  even number
71 – 38 = 33              odd number – even number                   odd number
72 – 37 = 35              even number – odd number                   odd number
71 – 37 = 34               odd number – odd number                   even number

72 x 38 = 2736             even number x even number                  even number
71 x 38 = 2698             odd number x even number                   even number
72 x 37 = 2664             even number x odd number                   even number
71 x 37 = 2627              odd number x odd number                   odd number

A multiple of a number is the product of that number with an integer multiplier. For example, 8
is a multiple of 2 and a multiple of 4 since 8 = 2 x 4 and both 2 and 4 are integers. Note that
even though 9 can also be expressed as the product of 4 and another number (9 = 2.25 x 4), it is
not a multiple of 4 since its multiplier is not an integer.
Whenever there is a multiple, there are factors. The number 18 is a multiple of 6 because 3 x 6 =
18, which means 18 is also a multiple of 3. Therefore, 3 and 6 are factors of 18.

Organized List

Factors of 24                                    Factors of 16
1                         24                     1                         16
2                         12                     2                          8
3                          8                     4                          4
4                          6

8 Factors of 24; 1, 2, 3, 4, 6, 8, 12, 24           5 Factors of 16: 1, 2, 4, 8, 16

Factors of 6                                     Factors of 1
1                         6                      1                         1
2                         3

4 Factors of 6: 1, 2, 3, 6                          1 Factor of 1: 1

Factors of 97
1                         97

2 Factors of 97

Determining Factors

Organizing lists, like those above, are one way of determining factors in a systematic fashion,
beginning with 1 and the number itself, and then 2 or the next possible factor and its factor
partner, and so on.

Another way to organize and display the factors of a number is using a factor rainbow.

Factor Rainbow
Concrete models and other pictorial models can also be used. For example, to determine the
factors of 12, take 12 square tiles and try to arrange them into a rectangle. Record the length and
width of each rectangle you can make; these are the factor pairs. This can also be done
pictorially by drawing rectangle on grid paper.

Concrete Models

1 x 12

2x6                                            3x4

Prime and Composite Numbers

Prime Numbers

If you are talking about whole numbers, a prime number has exactly two different factors, 1 and
itself. If you’re talking about integers, a prime number has no factors other than 1, -1, itself and
its opposite. Numbers with exactly two factors have a special name – prime numbers, or primes.
There are very small prime numbers such as 2, 3, and 5, but there are also large ones, such as
6299. The number 1 is not prime. Neither is -1. A common assumption is that 1 is a prime
number. It is really lonely number because it is neither prime nor composite! It’s not prime
because it does not have exactly two factors, and it’s not composite because it does not have
more than two factors.

Composite numbers

Every whole number except 1 is either a composite number or a prime number. Composite
numbers are like composite materials. They’re made of more than two different factors! Every
whole number other than 1 that is not a prime is called a composite number, and has three or
more factors. Note that 2 is the only even prime number and the number 1 is neither prime nor
composite 9mathematicians call it a unit).

Perfect Numbers

To Olympic divers and gymnasts, 10 is a perfect number. In mathematics, a perfect number is
equal to the sum of all its whole
Determining Whether a Number Is Prime

There are many ways to decide whether a number is prime. On way, which is very tedious, is to
try to divide the number by every possible smaller number to see how many factors it has. An
interesting way to find the prime number between 1 and 100 is to use a technique called the
Sieve of Eratosthenes.

Finding Prime Numbers from 1 to 100: The Sieve of Eratosthenes

Use a 100 chart and coloured counters.

Step 1: Place a blue counter on 1.

Step 2: Place red counters on every multiple of 2 but not 2 itself.

Step 3: Place yellow counters on every uncovered multiple of 3 but not 3 itself.

Step 4: Place blue counters on uncovered multiples of 5 but not 5 itself.

Step 5: Place green counters on uncovered multiples of 7 but not 7 itself.

The uncovered numbers are primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, and 97

Prime Factorization

The primes are the building blocks of our whole number system in the sense that each whole
number can be broken down into prime factors in one unique way. This is called prime
factorization; for example, 56 = 2 x 2 x 2 x 7.

You can use factor tress to determine these prime factors. There is often more than one factor
tree for any given number, but they all end up with the same list of prime factors.

56

7     8

7     2 2 2

56

2     28

2     14   2
2      7 2 2

Common Factors and GCF

Sometimes you need to determine factors that two numbers have in c0mmon. Common factors
can be used to solve many mathematical problems. They are also useful for expressing fractions
in lowest or simplest terms.

Factors of 18: 1, 2, 3, 6, 9, 18,

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Common Factors: 1, 2, 3, and 6

Greatest Common Factor (GCF): 6

Common Multiples and LCM

Multiples of a number are the products of the number and other factors. Sometimes you need to
determine the multiples that two numbers have in common. Common multiples come in handy
for solving problems, especially determining the least common denominator (the least common
multiple of the numbers in the denominators) with fractions. The smallest number (other than
zero) that is a multiple of two or whole numbers is the least common multiple (LCM).

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, …

Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, …

There are an infinite number of common multiples of 12 and 15: 60, 120, 180, …

Lowest Common Multiple (LCM): 60

Positive Exponents

Suppose you multiply by the same factor more than once. You can show the repeated factors
using exponential form, where the base is the repeated factor, and the exponent tells the number
of repetitions. Writing number using exponents is representing a number in the unit of the base.
For example, 8 = 2 x 2 x 2, so we write 8 = 2 3 . In effect, we are using number theory ideas since we
are thinking of 8 units of 2. We say that 8 is “ the third power of 2” and is 2 3 is “2 to the third.”

base      exponent
23
Power

3 is the exponent to which the base of 2 is raised

Using a power is shorthand for multiplying, just as using multiplication is shorthand for adding.
2 3 is quicker to write than 2 x 2 x 2, just as 3 x 2 is quicker to write than 2 + 2 + 2.

Repeated Factors            Write                         Say                        Standard Form

3x3                         32                            Three to the second        9
power or three squared
3x3x3                       33                            Three to the third power   27
or three cubed

There is an agreement among mathematicians that any number to the zero power is one. So, 40 = 1 and
620 = 1.

Negative Exponents

4 -1 = 1/41 = 1/4 When the base is 4, each time you decrease the exponent by 1, the value is ¼ as large.

4 -2 = 1/42 = 1/16

If the base were 25, each time you decreased the exponent by 1, the value would be 1/25 as large. So,
negative exponents turn whole numbers into fractions.

Square Roots

A square rug with sides of 6 metres has an area of 6 m x 6 m, or 36 m2. When a number is multiplied by
itself, the product is the square of the number.

Factor form: 6 x 6 = 36

Write: 62 = 3

Say: 6 squared is equal to 36 or 36 is the square of 6.

When the product of two identical factors is a second number, the factor is the square root of the
number.

Write: √36 = 6

Say: The square root of 36 is equal to 6.

If a number is not a perfect square, its square root is not an integer.
N                                   N2                                   √n

1                                     1                                  1.000
2                                     4                                  1.414
3                                     9                                  1.732
4                                    16                                  2.000
5                                    25                                  2.236
6                                    36                                  2.449
7                                    49                                  2.646
8                                    64                                  2.828
9                                    81                                  3.000
10                                  100                                  3.162

Irrational Roots

If a whole number is not a perfect square, its square root is irrational. This means it cannot be
represented s a ratio of integers. It is represented by a non-repeating decimal that does not end.

√2 = 1.4142135

Logarithms

Scientists use logarithms to measure the magnitude of earthquakes. A logarithm is an exponent. A
common logarithm (or log) is just an exponent of 10. So, if someone asks for the log of 1000, they’re
asking for the exponent that would go with 10 to have a value of 1000. In other words, 10 raised to what
power is equal to 1000? The answer is 3.

Multiplication and Division Principles

1. Multiplication and division “undo’ each other. They are related inverse operations, e.g.,
if 12 ÷ 3 = 4, then 3 x 4 = 12
2.    You can multiply numbers in any order (the commutative property). However, with
division, the order in which you divide the numbers matters.
3.    To multiply two numbers, you can divide one factor and multiply the other by the same
amount without changing product (the associative property), e.g., 8 x 3 = (8 ÷2) x (3 x
2) = 4 x 6
4.    To divide two numbers, you can multiply or divide both umbers by the same amount
without changing the quotient, e.g., 15÷ 3 = (15 x 2) ÷ (3 x 2) = 30 ÷ 6.
5.    You can multiply in parts (the distributive property), e.g., 5 x 4 = 3 x 4 + 2 x 4.
6.    You can multiply in parts by breaking up the multiplier, e.g., 6 x 5 = 2 x 3 x 5.
7.    You can divide in parts by splitting the dividend into parts, but not the divisor (the
distributive property), e.g., 48 ÷ 8 = 32 ÷ 8 + 16 ÷ 8.
8.    You can divide by breaking up the divisor, e.g., 36 ÷ 6 = 36 ÷ 3 ÷ 2.
9.    When you multiply by 0, the product is 0.
10.   When you divide 0 by any number but 0, the quotient is 0.
11. You cannot divide by 0.
12. When you multiply or divide a number by 1, the answer is the number you started with.

Divisibility Rules

A number is divisible by:         If:                                Test with 324

2                   The ones digit is 0, 2, 4, 6, 8324: 4, an even number, is in
(or, it is an even number)      the ones place. So, 324 is
divisible by 2
3                     The sum of the digits is    324: 3 + 2 + 4 = 9 is divisible
divisible by 3               by 3. So, 324 is too.
4                  The number formed by the last 324: 24 is divisible by 4. SO,
two digits is divisible by 4             324 is too.
5                      The last digit is 0 or 5    324: 4, the last digit, is not 0
or 5. So, 324 is not divisible
by 5.
6                   The number is divisible by 2 324: 324 is divisible by 2. 324
and 3               is divisible by 3. So, 324 is
divisible by 6.
9                     The sum of the digits is         324: 3 + 2 + 4 = 9 9 is
divisible by 9         divisible by 9. So, 324 is too
10                       The final digit is 0      324: 4, the final digit, is not 0.
So, 324 is not divisible by 10.

Divisibility tests for 7 and 8 are not as simple as the tests for the other numbers from 1 through
10. Just go ahead and do the division.

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