Proving the Property: Relatively Prime Factors and Division

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SUMMARY

The discussion centers on the mathematical property that if \( pq = ab \) where \( p \) and \( b \) are relatively prime, then \( p \) must be a factor of \( a \) and \( b \) must be a factor of \( q \). This conclusion is derived from the fact that since \( ab = pq \), \( p \) divides \( ab \). Given that \( (p, b) = 1 \), it follows that \( p \) must divide \( a \). The argument is symmetric for the other direction, confirming the relationship between the factors.

PREREQUISITES
  • Understanding of prime factorization
  • Knowledge of relatively prime numbers
  • Familiarity with basic algebraic manipulation
  • Concept of divisibility in integers
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If pq=ab where p, b are relatively prime, p must be a factor of a and b must be a factor of q.
 
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well, since ab = pq, p divides ab. but (p, b) = 1 and so p must divide a [this is obvious, trust me. if you don't see it, break them up into prime factors and deal with them case by case]. the other direction is symmetric.
 

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