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Prove a = b

  1. Apr 21, 2010 #1
    Show that if a [tex]\equiv[/tex] b mod p for all primes p, then a = b.
     
  2. jcsd
  3. Apr 21, 2010 #2
    Well, a - b must be divisible by all primes p. What is the only way for this to happen?
     
  4. Apr 21, 2010 #3
    Oh hmmm....

    The only way is if (a - b) is zero. How would I formally write this up? I guess a - b can't be the product of all primes???
     
  5. Apr 21, 2010 #4
    Every nonzero integer can only be divisible by a finite number of primes.
     
  6. Apr 21, 2010 #5

    CRGreathouse

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    In a sense, that's what 0 is. It's the "infinity" of the divisibility relation.
     
  7. Apr 22, 2010 #6

    HallsofIvy

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    If [itex]a> b[/itex] then a- b is a positive number. Since there are an infinite number of primes, there exist a prime, p> a- b. Then p cannot divide a- b so [itex]a\ne b (mod p)[/itex].

    If [itex]b> a[/itex] just use b- a instead of a- b.
     
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