How many triples?

Algebra Level 3

How many sets of (unordered) triples of integers a , b , c a,b,c are there to the following equation: a + b + c = a ⋅ b ⋅ c a + b + c = a \cdot b \cdot c

Note: Unordered means that, if 50 , 51 , 52 {50,51,52} were a solution, only count it once (and not 50 , 52 , 51 {50,52,51} etc.)

1 Infinitely many 2 Finitely many ( > 2 ) (> 2) 0

Stephen Mellor by Stephen Mellor , 20, United Kingdom

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1 solution

Sathvik Acharya
Dec 31, 2017

Assuming a = 0 a=0 , the equation a + b + c = a b c a+b+c=abc changes to b + c = 0 b+c=0 that is b = − c b=-c . Obviously there are infinitely many solutions for ( b , c ) (b,c) for example, ( − 1 , 1 ) , ( − 2 , 2 ) , ( − 3 , 3 ) (-1,1), (-2,2), (-3,3) and so on. Thus there are infinitely many solutions to the above equation a + b + c = a b c a+b+c=abc

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yep. Also note that 0,0,0 is a solution. Furthermore 1,2,3 is a solution with no zeroes

Stephen Mellor - 3 years, 5 months ago

-1, -2, -3 also is a solution with no zeroes

E Koh - 1 year, 4 months ago

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