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Question Number 213663 by efronzo1 last updated on 13/Nov/24
Given a,b,c is natural numbers such that (a−b)(b−c)(c−a)=a+b+c. find min value of a+b+c
Givena,b,cisnaturalnumberssuchthat(a−b)(b−c)(c−a)=a+b+c.findminvalueofa+b+c
Answered by Ghisom last updated on 13/Nov/24
let a<b<c ( _(c<a<b possible)^(also b<c<a or) ) b=a+p∧c=b+q p(p+q)q=3a+2p+q a=((p^2 q+pq^2 −2p−q)/3) ⇒ p=3m∧q=3n a=9m(m+n)n−2m−n b=9m(m+n)n+m−n c=9m(m+n)n+m+2n a+b+c=27m(m+n)n if 0∈N minimum at m=n=a=b=c=a+b+c=0 if 0∉N minimum at m=n=1 a=15 b=18 c=21 a+b+c=54
leta<b<c(c<a<bpossiblealsob<c<aor)b=a+p∧c=b+qp(p+q)q=3a+2p+qa=p2q+pq2−2p−q3⇒p=3m∧q=3na=9m(m+n)n−2m−nb=9m(m+n)n+m−nc=9m(m+n)n+m+2na+b+c=27m(m+n)nif0∈Nminimumatm=n=a=b=c=a+b+c=0if0∉Nminimumatm=n=1a=15b=18c=21a+b+c=54

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