### Instruction: Suppose that $a,b,$ and $c$ are positive integers satisfying $(a+b+c)^3 - a^3 - b^3 - c^3 = 150$. Find $a+b+c$.### Response: I notice that the left-hand side of the equation looks similar to the expansion of $(a+b+c)^3$, except that some terms are missing.
To see what those terms are, I write out the full expansion: $(a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3b^2c + 3ac^2 + 3bc^2 + 6abc$.
Now I can see that the terms that are missing are the ones that involve two or three different variables: $3a^2b + 3a^2c + 3ab^2 + 3b^2c + 3ac^2 + 3bc^2 + 6abc$.
I can factor out a 3 from these terms and get $3(a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2 + 2abc)$.
This means that the equation can be rewritten as $(a+b+c)^3 - a^3 - b^3 - c^3 = 3(a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2 + 2abc)$.
Since $a,b,$ and $c$ are positive integers, and the right-hand side is a multiple of 3, the left-hand side must also be a multiple of 3.
This implies that $150$ is a multiple of 3, which is false.
Therefore, there are no positive integers $a,b,$ and $c$ that satisfy the equation.
This means that $a+b+c$ is undefined.