1/a +1/b +1/c= abc
find values of
a
b
c
Answers
Answer:
Let's assume positive integers a,b,c such that c>=b>=a
The given equation becomes:
ab + bc + ac = abc….(*)
From (*) a or b or c not equal to zero.
By trying all positive integers, a=1,2,3,…
Put a=1 we have
b + c + bc= bc,==> b+c=0 (no solution for a=1)
Put a=2,we have:
2b+2c+bc=2bc ==> bc-2b-2c=0,this can be factorize to (b-2)(c-2)=4 which becomes
b-2=1 or c-2=4,==>b=3,c=6 and b-2=2 or c-2=2
==>b=c=4
Also put a=3 we have:
3b+3c+bc=3bc,==>2bc-3b-3c=0,can be factorize to (2b-3)(2c-3)=9
==> 2b-3=1 or 2c-3=9, b=2,c=6 and 2b-3=3 or 2c-3=3, b=c=3.
Similarly put a=4 we have:
4b+4c+bc=4bc,==> 3bc-4b-4c=0,in the same way, (3b-4)(3c-4)=16:
3b-4=1 or 3c-4=16,==>b=5/3,c=20/3 (not integers,no solution) also 3b-4=2 or 3c-4=8==>
b=2,c=4(solution but b>=a) and 3b-4=4 or 3c-4=4,==> b=8/3,c=8/3(not integers, so no solution). For a=4,5,6,… no solution.
Hence (a,b,c) = (2,3,6),(2,4,4),(3,3,3)
Step-by-step explanation:
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