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Suppose that a,b, and c are positive integers satisfying (a+b+c)^3 - a^3 - b^3 - c^3 = 150. 

Find a+b+c.

 

\(\small{ \begin{array}{|rcll|} \hline \boxed{(a+b+c)^3 - a^3 - b^3 - c^3 = 150} \\ \hline (a+b+c)^3 - a^3 - b^3 - c^3 &=& \Big( (a+b)+c \Big)^3 -a^3-b^3-c^3 \\ && \boxed{ ( (a+b)+c )^3 = (a+b)^3 + 3(a+b)^2c+3(a+b)c^2+c^3 \\= (a+b)^3+c^3+3c(a+b)(a+b+c) } \\ &=& (a+b)^3+c^3+3c(a+b)(a+b+c) -a^3-b^3-c^3 \\ && \boxed{ (a+b)^3 = a^3+3a^2b+3ab^2+b^3 \\ = a^3+b^3+3ab(a+b) } \\ &=& a^3+b^3+3ab(a+b)+c^3+3c(a+b)(a+b+c) -a^3-b^3-c^3 \\ &=& a^3+b^3+c^3+ 3ab(a+b)+3c(a+b)(a+b+c) -a^3-b^3-c^3 \\ &=& 3ab(a+b)+3c(a+b)(a+b+c) \\ &=& 3(a+b) \Big( ab+c(a+b+c) \Big) \\ &=& 3(a+b)(ab+ac+bc+c^2) \\ &=& 3(a+b)\Big( a(b+c)+c(b+c) \Big) \\ &=& 3(a+b)(b+c)(c+a) \\ \hline \boxed{3(a+b)(b+c)(c+a)=150} \\ \hline \end{array} } \)

 

\(\begin{array}{|lrcll|} \hline & 3(a+b)(b+c)(c+a) &=& 150 \quad & | \quad :3 \\ & (a+b)(b+c)(c+a)&=& 50 \quad & | \quad 50 = 2\cdot 5^2 \\ & (a+b)(b+c)(c+a)&=& 2\cdot 5^2 \\\\ (1)& 2\cdot 5 \cdot 5 &=& 2\cdot 5^2 \\ (2)& 5\cdot 2 \cdot 5 &=& 2\cdot 5^2 \\ (3)& 5\cdot 5 \cdot 2 &=& 2\cdot 5^2 \\\\ (1) & (a+b) &=& 2 \\ & (b+c) &=& 5 \\ & (c+a) &=& 5 \\ & (a+b) + (b+c) +(c+a) &=& 2 + 5 + 5 \\ & 2(a+b+c) &=& 12 \quad & | \quad :2 \\ & \mathbf{ (a+b+c) }& \mathbf{=} & \mathbf{ 6 } \\\\ (2) & (a+b) &=& 5 \\ & (b+c) &=& 2 \\ & (c+a) &=& 5 \\ & (a+b) + (b+c) +(c+a) &=& 5 + 2 + 5 \\ & 2(a+b+c) &=& 12 \quad & | \quad :2 \\ & \mathbf{ (a+b+c) }& \mathbf{=} & \mathbf{ 6 } \\\\ (3) & (a+b) &=& 5 \\ & (b+c) &=& 5 \\ & (c+a) &=& 2 \\ & (a+b) + (b+c) +(c+a) &=& 5 + 5 + 2 \\ & 2(a+b+c) &=& 12 \quad & | \quad :2 \\ & \mathbf{ (a+b+c) }& \mathbf{=} & \mathbf{ 6 } \\\\ \hline \Rightarrow &&& \{a=1, b=1, c=4\} \text{ or }\\ &&& \{a=4, b=1, c=1\} \text{ or } \\ &&& \{a=1, b=4, c=1\} \\ \hline \end{array}\)

 

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heureka Jul 20, 2018