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5.1

Number Theory

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Number Theory

The study of numbers and their properties. The numbers we use to count are called natural

numbers, N, or counting numbers.

{1,2,3,4,5,...}N

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Factors

The natural numbers that are multiplied together to equal another natural number are called factors of the product.

Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

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Divisors

If a and b are natural numbers and the quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.

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Prime and Composite Numbers

A prime number is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1.

A composite number is a natural number that is divisible by a number other than itself and 1.

The number 1 is neither prime nor composite, it is called a unit.

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Rules of Divisibility

285The number ends in 0 or 5.

5

844 since 44 4

The number formed by the last two digits of the number is divisible by 4.

4

846 since 8 + 4 + 6 = 18

The sum of the digits of the number is divisible by 3.

3846The number is even.2

ExampleTestDivisible by

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Create a list from 1 – 100

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

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The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a unique product of prime numbers.

This unique product is referred to as the prime factorization of the number.

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Finding Prime Factorizations

Branching Method: Select any two numbers whose product is

the number to be factored. If the factors are not prime numbers,

continue factoring each number until all numbers are prime.

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Example of branching method

Therefore, the prime factorization of

3190 = 2 • 5 • 11 • 29.

3190

319 10

11 29 2 5

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1. Divide the given number by the smallest prime number by which it is divisible.

2. Place the quotient under the given number.3. Divide the quotient by the smallest prime

number by which it is divisible and again record the quotient.

4. Repeat this process until the quotient is a prime number.

Division Method

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Write the prime factorization of 663.

The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17

Example of division method

13

3

17

221

663

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Example 1: p. 218# 37

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Greatest Common Factor

The greatest common factor (GCF) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.

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Finding the GCF of Two or More Numbers Determine the prime factorization of each

number. List each prime factor with smallest

exponent that appears in each of the prime factorizations.

Determine the product of the factors found in step 2.

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Example (GCF)

Find the GCF of 63 and 105. 63 = 32 • 7 105 = 3 • 5 • 7

Smallest exponent of each factor:3 and 7

So, the GCF is 3 • 7 = 21.

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Find the GCF between 36 and 54

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Least Common Multiple

The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.

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Finding the LCM of Two or More Numbers Determine the prime factorization of each

number. List each prime factor with the greatest

exponent that appears in any of the prime factorizations.

Determine the product of the factors found in step 2.

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Example (LCM)

Find the LCM of 63 and 105. 63 = 32 • 7105 = 3 • 5 • 7

Greatest exponent of each factor:32, 5 and 7

So, the LCM is 32 • 5 • 7 = 315.

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Find the LCM between 36 and 54

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Example of GCF and LCM

Find the GCF and LCM of 48 and 54. Prime factorizations of each:

48 = 2 • 2 • 2 • 2 • 3 = 24 • 354 = 2 • 3 • 3 • 3 = 2 • 33

GCF = 2 • 3 = 6

LCM = 24 • 33 = 432

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Homework

P. 218# 15 – 54 (x3)