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1.11: Prime Numbers
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.11%3A_Prime_Numbers
This gives us a slight shortcut to finding primes with the Sieve of Eratosthenes: in our example above, once we have circled 7 and crossed out its multiples in the example above, every other number cu...This gives us a slight shortcut to finding primes with the Sieve of Eratosthenes: in our example above, once we have circled 7 and crossed out its multiples in the example above, every other number currently in the list that has not yet been circled or crossed out is guaranteed to be prime and can immediately be circled, since 7 is the largest prime number that is less than or equal to the square root of 100.
2.3: Predicate Logic
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/02%3A_Logic/2.03%3A_Predicate_Logic
This page covers the use of predicates and quantifiers in logic for effective communication, including negation and application through exercises. It provides examples involving bees, flowers, trees, ...This page covers the use of predicates and quantifiers in logic for effective communication, including negation and application through exercises. It provides examples involving bees, flowers, trees, and moose, highlighting logical equivalences and implications related to these subjects. The text encourages readers to translate natural language statements into logical notation and practice negation and interpretation, reinforcing their understanding of predicate logic in various scenarios.
2: Prime Numbers
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/02%3A_Prime_Numbers
Prime numbers, the building blocks of integers, have been studied extensively over the centuries. Being able to present an integer uniquely as product of primes is the main reason behind the whole the...Prime numbers, the building blocks of integers, have been studied extensively over the centuries. Being able to present an integer uniquely as product of primes is the main reason behind the whole theory of numbers and behind the interesting results in this theory. Many interesting theorems, applications and conjectures have been formulated based on the properties of prime numbers.
2.8: Find Multiples and Factors (Part 2)
https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_1e_(OpenStax)/02%3A_Introduction_to_the_Language_of_Algebra/2.08%3A_Find_Multiples_and_Factors_(Part_2)
A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each ...A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
5.2: Number Theory
https://math.libretexts.org/Courses/College_of_the_Canyons/Math_130%3A_Math_for_Elementary_School_Teachers_(Lagusker)/05%3A_Number_Theory/5.02%3A_Number_Theory
Let $mn = p$ , then $m$ and $n$ are factors of $p$ and $p$ is a multiple of $m$ and $n$ Factors are always smaller than the given number, whereas multiples are always bigger than the give...Let $mn = p$ , then $m$ and $n$ are factors of $p$ and $p$ is a multiple of $m$ and $n$ Factors are always smaller than the given number, whereas multiples are always bigger than the given number. List all the factors and the first four multiples of 30. Cross out 0 and 1 (neither prime nor composite) and circle 2 (the first prime) Circle 3 (prime) and cross out all multiples of 3. Categorize the following as Prime, Composite or Neither: 0, 1, 2, and any negative number
2.8: Find Multiples and Factors (Part 2)
https://math.libretexts.org/Courses/Grayson_College/Prealgebra/Book%3A_Prealgebra_(OpenStax)/02%3A_Introduction_to_the_Language_of_Algebra/2.08%3A_Find_Multiples_and_Factors_(Part_2)
A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each ...A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
4.7: Find Multiples and Factors (Part 2)
https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/04%3A_Fractions/4.07%3A_Find_Multiples_and_Factors_(Part_2)
A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each ...A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
6.1: Prime numbers
https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/6%3A_Prime_numbers/6.1%3A_Prime_numbers
Then $n$ is called a prime number if $n$ has exactly two positive divisors, $1$ and $n.$ Then $q$ is also a prime divisor of $n$ and $q < m < \sqrt{n} < p.$ This is a contradiction. B...Then $n$ is called a prime number if $n$ has exactly two positive divisors, $1$ and $n.$ Then $q$ is also a prime divisor of $n$ and $q < m < \sqrt{n} < p.$ This is a contradiction. But this is impossible since there is no prime that divides 1 and as a result $q$ is not one of the primes listed. Consider the sequence of integers $(n+1)!+2, (n+1)!+3,...,(n+1)!+n, (n+1)!+(n+1)$
10.8: Find Multiples and Factors (Part 2)
https://math.libretexts.org/Courses/Coastline_College/Math_Concurrent_Support_(Tran)/10%3A_Introduction_to_the_Language_of_Algebra/10.08%3A_Find_Multiples_and_Factors_(Part_2)
A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each ...A prime number is a counting number greater than 1 whose only factors are 1 and itself. A composite number is a counting number that is not prime. To determine if a number is prime, divide it by each of the primes, in order, to see if it is a factor of the number. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
8.2: Prime Numbers and Prime Factorizations
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/08%3A_Topics_in_Number_Theory/8.02%3A_Prime_Numbers_and_Prime_Factorizations
We showed how the Euclidean Algorithm can be used to find the greatest common divisor of two integers, $a$ and $b$ , and also showed how to use the results of the Euclidean Algorithm to write the g...We showed how the Euclidean Algorithm can be used to find the greatest common divisor of two integers, $a$ and $b$ , and also showed how to use the results of the Euclidean Algorithm to write the greatest common divisor of $a$ and $b$ as a linear combination of $a$ and $b$ .
8.2: Prime Numbers and Prime Factorizations
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/08%3A_Topics_in_Number_Theory/8.02%3A_Prime_Numbers_and_Prime_Factorizations
We showed how the Euclidean Algorithm can be used to find the greatest common divisor of two integers, $a$ and $b$ , and also showed how to use the results of the Euclidean Algorithm to write the g...We showed how the Euclidean Algorithm can be used to find the greatest common divisor of two integers, $a$ and $b$ , and also showed how to use the results of the Euclidean Algorithm to write the greatest common divisor of $a$ and $b$ as a linear combination of $a$ and $b$ .

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