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1. Fundamental Theorem Of Arithmetic

   Linked via "greatest common divisor (GCD)"

   If $p1$ divides $n$, then $p1$ must divide the right-hand product. By repeated application of Euclid's Lemma, $p1$ must equal some $qj$. We can cancel this common factor and proceed recursively on the remaining factors until all factors are exhausted, demonstrating that the sets of primes $\{pi\}$ and $\{qi\}$ must be identical, along with their respective counts (exponents).
   A related, though less efficient, method for proving uniqueness involves the concept of the **[greatest common divisor (GCD)](/entries/gr…