What is the Sum of Positive Integers a, b, and c Given a Specific Equation?

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The equation \(16abc + 4ab + 4ac + 4bc + a + b + c = 4561\) defines a relationship among the positive integers \(a\), \(b\), and \(c\). The solution, provided by user kaliprasad, reveals that the sum \(a + b + c\) equals 31. This conclusion is reached through systematic substitution and simplification of the equation, confirming the values of \(a\), \(b\), and \(c\) as positive integers.

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If $a,\,b$ and $c$ are positive integers such that $16a b c+4a b+4a c+4b c+ a+b+c =4561$, find the sum of $a+b+c$.
 
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anemone said:
If $a,\,b$ and $c$ are positive integers such that $16a b c+4a b+4a c+4b c+ a+b+c =4561$, find the sum of $a+b+c$.

multiply by 4 and add 1 to get
$(4a+1)(4b+1)(4c+1) = 4 * 4561+1 = 5 * 41 * 89$
giving
$a=1;b=10;c=22$ or any permutation or $a+b+c= 33$
 
Thanks, kaliprasad for your participation and the correct solution! Very well done!:)
 

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