Prime numbers are natural numbers greater than 1 that are only divisible by themselves and 1. This text delves into prime factorization, a method to break down composite numbers into prime components. It discusses the division and factor tree methods, and practical applications like calculating GCD and LCM. Understanding prime factorization is crucial for solving complex mathematical problems and exploring number relationships.
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Natural numbers greater than 1 that are divisible by only two distinct positive divisors: 1 and themselves
Notable Examples
Examples include 2, 3, 5, 7, 11, and 13
The study of prime numbers has its roots in antiquity, with Euclid of Alexandria demonstrating their infinitude around 300 B.C
The process of breaking down a composite number into a product of prime numbers
Division Method
Involves dividing the number by the smallest prime number that yields a whole number quotient and repeating this process until reaching 1
Factor Tree Method
Involves systematically breaking down each composite factor into smaller factors until only primes remain
Examples include the factorization of 14 into 2 x 7 and 999 into 3^3 x 37
The GCD is the product of the lowest powers of common prime factors
The LCM is the product of the highest powers of all prime factors present in either number
Prime factorization aids in calculating the total number of divisors a number has by incrementing each exponent in its prime factorization by one and then multiplying these figures together
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Definition of prime numbers
Natural numbers > 1, divisible only by 1 and themselves.
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Exceptional prime number
2 is the only even prime number.
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Prime vs Composite numbers
Primes have no divisors other than 1 and themselves; composites have additional divisors.
