1. **State the problem:** We have a radioactive isotope with an initial mass of 250 grams that halves every 10 years. We want to find the remaining mass after 6 years. 2. **Formula used:** The mass remaining after time $t$ for a substance with half-life $T$ is given by: $$m(t) = m_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}$$ where $m_0$ is the initial mass, $t$ is the elapsed time, and $T$ is the half-life. 3. **Apply the values:** Here, $m_0 = 250$, $t = 6$, and $T = 10$. $$m(6) = 250 \times \left(\frac{1}{2}\right)^{\frac{6}{10}}$$ 4. **Calculate the exponent:** $$\frac{6}{10} = 0.6$$ 5. **Evaluate the power:** $$\left(\frac{1}{2}\right)^{0.6} = 2^{-0.6}$$ Using a calculator or logarithms, this is approximately 0.6598. 6. **Calculate the remaining mass:** $$m(6) = 250 \times 0.6598 = 164.95$$ 7. **Round to the nearest whole number:** $$\boxed{165}$$ grams remain after 6 years.