NÚMEROS PRIMOS: Número Primo ou Composto | Matemática Básica - Aula 20

Understanding Prime and Composite Numbers

Introduction to Prime Numbers

• The session begins with an introduction to the topic of prime numbers, emphasizing their definition and how to determine if a number is prime or composite.

• A prime number is defined as a natural number that has exactly two distinct natural divisors: 1 and itself.

Characteristics of Natural Numbers

• The discussion focuses on natural numbers, which include 0, 1, 2, 3, etc. It clarifies that:

• 0 is not a prime number because it does not meet the divisor criteria.

• 1 is also not considered a prime number since it only has one distinct divisor (itself).

Identifying Prime Numbers

• The number 2 is highlighted as the only even prime number; all other primes are odd.

• There are infinitely many prime numbers. Examples up to 59 are provided for memorization purposes.

• Gaps between primes are noted, with examples of missing even numbers like 4, 6, and odd numbers like 15 and 21 being discussed.

Definition of Composite Numbers

• A composite number is defined as a natural number that has more than two distinct natural divisors.

• Examples of composite numbers include:

• Even numbers greater than 2 (e.g., 4, 6, 8, 10)

• Odd composites such as 15, which can be divided by multiple factors (1, itself, and others).

Classification of Natural Numbers

• All natural numbers greater than or equal to 2 are classified either as primes or composites; they cannot be both simultaneously.

Method for Identifying Primality

• To determine if a natural number is prime:

• Perform successive divisions by known primes until the divisor exceeds the square root of the target number.

• If any division results in zero remainder before reaching this point, the number is composite.

Example Calculation

Understanding Prime Numbers and Divisibility

Testing the Divisibility of 253

• The number 253 is tested for divisibility by 2, concluding it is not divisible since it does not meet the criteria.

• Next, the sum of the digits (2 + 5 + 3 = 10) is calculated to check divisibility by 3; since 10 is not divisible by 3, neither is 253.

• When testing for divisibility by 5, it's noted that since 253 does not end in either a '0' or '5', it cannot be divisible by this number.

• The next prime number tested is 7. A long division shows that there’s a remainder when dividing, confirming that 253 isn't divisible by seven.

• Finally, when dividing by 11, the calculation reveals no remainder; thus, it confirms that 11 is a divisor of 253 and concludes that it is a composite number.

Exploring Another Candidate: The Number 223

• Moving on to test if the number 223 is prime, initial checks show it's not divisible by two as it's an odd number.

• For divisibility by three, summing its digits (2 + 2 + 3 =7), which isn’t divisible by three indicates that neither can be.

• Checking against five again shows no divisibility as it doesn’t end in '0' or '5'.

• When attempting division with seven through long division calculations results in a non-zero remainder indicating non-divisibility.

• Further tests with eleven also yield a non-zero remainder after performing long division.

Continuing Tests Until Confirmation

• As further divisions are attempted with thirteen and seventeen respectively, both result in non-zero remainders confirming they are also not divisors of the candidate number.

• After testing up to seventeen without finding any factors other than one and itself confirms that indeed, 223 remains undivided and thus qualifies as a prime number.

Conclusion on Prime Identification Process

• The process concludes once potential divisors exceed the square root of the candidate. In this case, testing stops at seventeen for 223 because further attempts would be unnecessary given prior results.