Find the smallest possible sum!

Number Theory Level 3

If a , b , c a,b,c are all positive integers satisfying a b c = 7 ( a + b + c ) abc= 7(a+b+c) , what is the smallest possible value of a + b + c ? a+b+c ?


The answer is 15.

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Vijay Simha by Vijay Simha , 47, USA

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1 solution

Vijay Simha
Mar 4, 2017

Without loss of generality, since one of the integers must be divisible by 7. Call this one a. For the smallest possible value, we should try starting with a = 7. Then we have bc = 7 + b + c, which yields 8 = (b − 1)(c − 1).

Then {b, c} = {3, 5} or {2, 9}.

Hence {a, b, c} = {7, 3, 5} yields the smallest sum

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Somewhat more interestingly, are there many other solutions? If yes, how can we find them?

Calvin Lin Staff - 4 years, 3 months ago

You didn't write that a , b , c a,b,c are non-zero, so i assumed that ( a , b , c ) = ( 0 , 0 , 0 ) (a,b,c) = (0,0,0)

Fidel Simanjuntak - 4 years, 3 months ago

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The problem states that they are positive integers, hence non-zero.

Calvin Lin Staff - 4 years, 3 months ago

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Ah, i didn't realize it.. Thanks

Fidel Simanjuntak - 4 years, 3 months ago

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