Question

unique factorization theorem (fundamental theorem of arithmetic)

Answer 1

In number theory, 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 prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.[4][5][6] For example,

1200 = 24 × 31 × 52 = 5 × 2 × 5 × 2 × 3 × 2 × 2 = ...

The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second, no matter how this is done, there will always be 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 (e.g., 12 = 2 × 6 = 3 × 4).

This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2×1 = 2×1×1 = ...

Answer 2

Answer:

Step-by-step explanation:

FUNDAMENTAL THEOREM OF ARITHMETIC :  

According to the fundamental theorem of arithmetic every composite number can be written or  factorized as the product of primes and this factorization is unique, apart from the order in which the prime factors occur.  

Fundamental theorem of arithmetic , is also called, UNIQUE FACTORIZATION THEOREM.

Composite number =  product of prime numbers

Or  

Any integer greater than one, either be a prime number or can be written as a product of prime factors.

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