How can you find all parameter estimates are statistically significant but without performing any sophisticated calculations?

Normally an exam question like that can be solved by simply calculating the $t$-statistics which can be simply done as:

$$t_{hat{beta}} = frac{hat{beta}-beta_0}{se(hat{beta})}$$

where $hat{beta}$ is the estimate, $beta_0$ is the assumed value of beta under null (usually 0) and $se(hat{beta})$ standard error of the coefficient and then compare it to $5%$ critical value (for large $n$ and two sided hypothesis approximately $1.96$).

However, you should ask your teacher/supervisor to clarify what is considered 'sophisticated calculation' - some might consider even subtraction and division sophisticated. An approximate rule of thumb that requires even less calculation is to see if the coefficient $hat{beta}$ (and assuming null hypothesis assumes $beta_0=0$) is twice as large as the standard error.