23. Prime numbers and the fundamental theorem of arithmetic

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The Prime Numbers are one of the fundamental concepts of mathematics. A prime number is a natural number greater than 1 that has only two distinct natural divisors: itself and 1. In other words, a number is prime if it cannot be formed by multiplying two smaller numbers together. The first prime numbers are 2, 3, 5, 7, 11, and so on. The number 1 is not considered a prime number because it has only one divisor.

The importance of prime numbers in mathematics cannot be overemphasized. They are the "building blocks" of the natural numbers, and thanks to the Fundamental Theorem of Arithmetic, we know that every natural number greater than 1 can be written as a product of prime numbers, and this representation is unique , except for the order of the prime factors. This is a powerful and surprising fact that allows us to decompose any number into its "prime parts".

For example, the number 315 can be decomposed into prime numbers like 3*3*5*7. No other set of primes multiplied together will give 315, so this is the only way to break 315 into primes.

To find the prime factors of a number, we start by dividing the number by the smallest prime number, which is 2. If the number is divisible by 2, then 2 is a prime factor. We keep dividing the number by 2 until it is no longer divisible. Then we move on to the next prime number, which is 3, and repeat the process. We continue in this way until the remaining number is a prime number.

The ability to decompose numbers into prime factors is useful in many areas of mathematics, including solving Diophantine equations, public-key cryptography, and number theory.

The Fundamental Theorem of Arithmetic was first proved by Euclid over two millennia ago. It states that every integer has a unique decomposition into prime factors. This means that while there may be many different ways to decompose a number into factors, only one of those decompositions will consist entirely of prime numbers, and that decomposition will be the same no matter how you find it.

For example, the number 24 can be factored in several ways: 1*24, 2*12, 3*8, 4*6. However, only one of these decompositions is the prime factorization: 2*2*2*3. No matter how you factor 24, you'll always arrive at that same prime factorization.

In short, prime numbers are the building blocks of natural numbers. They are fundamental to our understanding of numbers and have many practical applications. The Fundamental Theorem of Arithmetic, which states that every number has a unique decomposition into prime factors, is one of the most profound and important truths in mathematics.

For the ENEM test, it is important to be familiar with the concepts of prime numbers and decomposition into prime factors. Questions about these topics are common and can be resolved quickly if you understand these concepts well.

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2424. Solving equations and inequalities