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If p_n assigned a positive exponent to any prime c while p_m assigned c a zero exponent, then the product of p_n would be congruent to 0 (mod c), but the product of p_m would not, and therefore the two products would not equal the same number.

This is basically just a restatement of the Fundamental Theorem of Arithmetic. See Gower's Answer 3 in his post.

Nope, it's actually a restatement of a lemma traditionally used to prove the Fundamental Theorem of Arithmetic, and quite a nice one too in my opinion. (It's "obviously" true in the sense that any exception feels like it'd violate basic behaviour we'd expect from modulo arithmetic on primes, and with a bit of head-scratching it even seems to be possible to prove that.)

No, it's not, and I specifically quoted the part of Gower's Answer 3 that mentions it in my original comment.

If you think this is a restatement of the fundamental theorem of arithmetic, maybe you think of the FTA as obvious after all. ;p

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