Fundamental theorem of arithmetic :-

Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique . apart from the other in which factors occur.

The fundamental theorem of arithmetic or the unique factorisation theorem would fail.

The prime factorisation theorem is also known as the fundamental theorem of arithmetic. So in that context, it does.

Because otherwise the fundamental theorem of arithmetic, the unique factorisation theorem, would fail.