Is a Prime Number Always a Factor of a Product of Integers?

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SUMMARY

A prime number p is a factor of a non-zero product of integers a*b if and only if it is a factor of a and/or b. This conclusion is supported by the fundamental theorem of arithmetic, which states that every integer has a unique prime decomposition. Therefore, if p divides the product a*b, it must also divide at least one of the integers a or b, as the prime decomposition of a*b is derived from the decompositions of a and b. The proof utilizes Euclid's lemma, confirming the relationship between prime factors and their products.

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How to proof?
A prime number p is a factor of a non-zero product of integers a*b if and only if it is a facotr of a and/or b.
 
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The "if" direction is obvious. For the "only if", you need to use the fundamental theorem of arithmetic. Because a*b has a unique prime decomposition, if a prime p divides a*b then it must be one of the primes in the decomposition. The prime decomposition of a*b is just the product of the decompositions of a and b, so p must divide a and/or b.

This sounds like a homework question! In which case you will need to be a lot more rigorous when you write it out.
 
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One side is trivial: if p is a factor of a and/or b, then it is clearly a factor of a * b.

For the converse implication, the easiest way I can think of is using the unique decomposition of any integer into its prime factors.
 
thx,guys
 
Caution. Some proofs of unique factorization may use this property of primes. And then it would be illegitimate to use unique factorization in the proof of this...
 
assume p is a factor of ab. then p|ab. since p is prime, euclid's lemma says p|a and/or p|b, thus p is a factor of a and/or b.
 

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