anonymous
  • anonymous
help! It said the Fundamental Theorem of Arithmetic is used to prove. Let m = p1^e1 * p2^e2 ... ps^es, where pi is a prime. m|n if and only if pi^ei | n for all i.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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dan815
  • dan815
all numbers can be written as a product of primes
dan815
  • dan815
primes have factors of 1 and themself
dan815
  • dan815
now this means that if m divide n then every prime^exponent divides n

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dan815
  • dan815
does it make sense?
dan815
  • dan815
maybe u need to say this statement too if m|n then n=k*m there fore k*(p1^e1*p2^e2....)
anonymous
  • anonymous
ok, m | n means mk = n for some integer k p1^e1 ( p2^e2 ... ps^es * k) = n implies p1^e1 | n p2^e2 * (p1^e2 * p3^e3 ... ps^es k) = n implies p2^e2 | n and so on to ps^es. How do you prove the other direction?
dan815
  • dan815
what do u mean
anonymous
  • anonymous
it's an if and only if statement
dan815
  • dan815
okay since u saw that n=k*m then n=k*p1^e1*p2^e2... therefore p1^e1,p2^e2... all have to be factors
dan815
  • dan815
and u say primes cannot be decomposed into other primes so u are done
dan815
  • dan815
there is no other prime representation for m, so it goes both ways
anonymous
  • anonymous
well, we just proved that direction. The other direction is if pi^ei | n for all i, then m|n
dan815
  • dan815
hmm to me its the same thing lol
dan815
  • dan815
ok how about saying it like this
dan815
  • dan815
if pi^ei | n for all i then (p1^e1)(p2^e2)(p3^e3).....(pn^en) | n so m|n
dan815
  • dan815
because if* p1|n and p2|n then p1*p2|n if p1 and p2 are prime
dan815
  • dan815
this has to be true as a 2 different primes cannot share factors
anonymous
  • anonymous
so a lemma was used a long the way Given p1 and p2 are primes. If p1| and p2|n, then p1*p2 | n
anonymous
  • anonymous
I think m * gcd(k1, k2, ... ks) = n will prove the result.

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