The Fundamental Theorem of Arithmetic states that 'every whole number greater than 1 is either a prime number or it can be expressed as a unique product of its prime factors', where 'unique product' means that there is only one product (where the order of the prime factors does not matter). If this theorem is false, many useful mathematical results will also be false.
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Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. (In modern terminology: every integer greater than one is divided evenly by some prime number.)
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