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Question Number 121219 by ZiYangLee last updated on 06/Nov/20
Given that a,b and c are three consecutive numbers, where a>b>c, such that its product is the same as its sum, that is abc=a+b+c, how many such (a,b,c)?
Giventhata,bandcarethreeconsecutivenumbers,wherea>b>c,suchthatitsproductisthesameasitssum,thatisabc=a+b+c,howmanysuch(a,b,c)?
Commented by liberty last updated on 06/Nov/20
i think it′s ambiguos. abc= a×b×c or a×100+b×10+c ?
ithinkit′sambiguos.abc=a×b×cora×100+b×10+c?
Answered by MJS_new last updated on 06/Nov/20
consecutive numbers means a, b, c ∈Z ⇒ b=a−1∧c=a−2 a(a−1)(a−2)=a+(a−1)+(a−2) a(a−1)(a−2)=3(a−1) (a−1)(a^2 −2a−3)=0 (a−3)(a−1)(a+1)=0 ((a),(b),(c) ) ∈{ (((−1)),((−2)),((−3)) ) , ((1),(0),((−1)) ) , ((3),(2),(1) )} three triples
consecutivenumbersmeansa,b,c∈Z⇒b=a−1∧c=a−2a(a−1)(a−2)=a+(a−1)+(a−2)a(a−1)(a−2)=3(a−1)(a−1)(a2−2a−3)=0(a−3)(a−1)(a+1)=0(abc)∈{(−1−2−3),(10−1),(321)}threetriples

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