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iran mathematical olympiad

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iran mathematical olympiad

Posted: Sat Sep 24, 2011 12:05 am
by mahathir
if $a,b,c,d$ are natural numbers and $ab=cd$,then,prove that,$a+b+c+d$ cannot be a prime.

Re: iran mathematical olympiad

Posted: Sun Sep 25, 2011 7:22 pm
by Masum
Try to prove the general one.
If $ab=cd$, prove that $a^n+b^n+c^n+d^n$ is not prime.

Re: iran mathematical olympiad

Posted: Sun Sep 25, 2011 9:37 pm
by nayel
The following method works for many problems of similar sort.
Let $g=(a,c), a=ga', c=gc'$. Then $a'b=c'd$ so $a'\mid c'd$. But $(a',c')=1$, hence $a'\mid d$. Let $d=ha'$. Then $a'b=c'ha'$ so $b=hc'$. Thus $a^n+b^n+c^n+d^n=(ga')^n+(hc')^n+(gc')^n+(ha')^n=(g^n+h^n)(a'^n+c'^n)$.
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