please show all work Prove if p is prime and x1 X2 xn are in

please show all work

Prove: if p is prime, and x_1, X_2,..., x_n are integers such that p|x_1|x_2 ... x_n, then p|x_k for some 1 lessthanorequalto k lessthanorequalto n. Use (3)(a) to prove the Fundamental Theorem of Arithmetic.

Solution

a) Since p is prime if p divides x1x2....xn

p has to divide atleast one otherwise say p does not divide xi individually for any i.

Then x1x2x3.....xn is divisible by p means p is of the form ab where a divides xi and b divides xj.

But this is a contradiction to p is prime

So p divides x1x2...xn means p divides atleast one xk for some k

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Fundamental theorem of Arithmetic:

This theorem states that every integer >1 is prime or product of prime factors.

By the part a proof, a number which is a product of only itself and 1 is prime.

If a number can be expressed as product of integers, then it can be expresed as product of only prime factors.

Hence proved