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

[anemone]

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 $a+b+c$.

 

[kaliprasad]

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 $a+b+c$.

Spoiler

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 $a+b+c= 33$

 

[anemone]

Thanks, kaliprasad for your participation and the correct solution! Very well done!:)