Mathematics I / Primes and the Fundamental Theorem of Arithmetic
Practice question · True or false

If 1 were counted as prime, unique factorisation would fail.

Hints
  1. How many ways could you then write 12 as a product of primes?
  2. 1 can be multiplied in as often as you like.
Show the answer

True

Why

True. 12=22⋅3=1⋅22⋅3=12⋅22⋅312 = 2^2 \cdot 3 = 1 \cdot 2^2 \cdot 3 = 1^{2} \cdot 2^{2} \cdot 3, factorisations would stop being unique, and the Fundamental Theorem of Arithmetic would need an awkward exception. Excluding 1 is what keeps 'exactly one factorisation' true as stated.

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