Question

Fundamental Theorem of Arithmetic
i) state that every negative integers is either prime or composite numbers can be expressed as the product of primes.
ii) has the basic importance in the development of number theory. iii) both i) & ii) are correct. iv) None

Answer 1

Answer:

the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1[3] either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.[4][5][6] For example,

The unique factorization theorem was proved by Gauss with his 1801 book Disquisitiones Arithmeticae.[1] In this book, Gauss used the fundamental theorem for proving the law of quadratic reciprocity.[2]

{\displaystyle 1200=2^{4}\cdot 3\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }{\displaystyle 1200=2^{4}\cdot 3\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }

The theorem says two things for this example: first, that 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.

The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, {\displaystyle 12=2\cdot 6=3\cdot 4}{\displaystyle 12=2\cdot 6=3\cdot 4}

Step-by-step explanation:

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