MHB 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 is presented, seeking the sum of the positive integers a, b, and c. Participants discuss methods to simplify and solve the equation, ultimately arriving at the correct values for a, b, and c. The solution reveals that the sum a + b + c equals a specific integer. The discussion highlights the importance of collaboration and problem-solving in mathematics. The final answer is derived through careful analysis and calculation.
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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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