imc

By Vieta's Formulas, we know that $a$ is the sum of the three roots of the polynomial $x^3-a{x}^2+bx-2010$. Again Vieta's Formulas tell us that $2010$ is the product of the three integer roots. Also, $2010$ factors into $2\cdot 3\cdot 5\cdot 67$. But, since the polynomial has only three roots, two of the four prime factors must be multiplied so that we are left with three roots. To minimize $a$, $2$ and $3$ should be multiplied, which means $a$ will be $6+5+67=78$ and the answer is $\boxed{\text{(A)78}}$. Note: If you are feeling unconfident about $78$, you can try to expand the expression $(x-5)(x-6)(x-67)$ which has roots $5$, $6$, and $67$. $$(x-5)(x-6)(x-67)=(x^2-11x+30)(x-67)$$ $$=x\left(x^2-11x+30\right)-67\left(x^2-11x+30\right)=x^3-11x^2+30x-67x^2+737x-2010=x^3-78x^2+767x-2010$$ As we can see, $a=78$, and since this is the least answer choice, we can be confident that the right option is $\boxed{78}$.