Saturday, 17 March 2018

elementary number theory - Does this follow from congruence?



Let a and b be two distinct prime numbers and x and y are integers. Is the following true?



(xymoda) and (xymodb). So, a|(xy) and b|(xy). This means xy=abϕ with ϕZ.



Can I state the last part or do I need to prove that somehow? It seems logical enough to me but it might be wrong. Your help is appreciated.



Answer



Let xy=am=bn, m,nZ.



am=bn



Since a and b are coprime and abn, an. Let n=aϕ.


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