Question

Fundamental theorem of arithmetic is also called as _________factorization theorem.

A)Algebra
B)Ambiguous
C)Unique
D)None of these

Answer 1

Hint: In number theory, the fundamental theorem of arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, (except for) the order of the factors.

 Complete step by step answer:
Now we will understand the theorem using an example,
1200 = 24 × 31 × 52 = 2 × 2 × 2 × 2 × 3 × 5 × 5 = 5 × 2 × 5 × 2 × 3 × 2 × 2 = ...

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 (e.g., 12 = 2 × 6 = 3 × 4).
This theorem is one of the main reasons why 1 is not considered as a prime number. If 1 was a prime number, then factorization into primes would not be unique; for example, 2 = 2 × 1 = 2 × 1 × 1 = ...
Which implies, this theorem is unique.
So this theorem is also called a Unique factorization theorem.
The correct answer is Option C.

Note:
Make sure that you are not confused with the fundamental theorem of algebra. Go through the Fundamental theorem of algebra also so that you don’t make mistakes any further. Also, check the calculations while you solve problems related to prime factorization theorem