1. **State the problem:** We need to find the prime factorization of 208 in the form $2^a \times b$, where $a$ and $b$ are integers and $b$ is not divisible by 2. 2. **Recall the prime factorization method:** To find the prime factors, we repeatedly divide the number by the smallest prime (starting with 2) until it is no longer divisible. 3. **Start dividing 208 by 2:** $$208 \div 2 = 104$$ 4. **Divide 104 by 2:** $$104 \div 2 = 52$$ 5. **Divide 52 by 2:** $$52 \div 2 = 26$$ 6. **Divide 26 by 2:** $$26 \div 2 = 13$$ 7. **Check if 13 is divisible by 2:** It is not, so we stop here. 8. **Count the number of times 2 divides 208:** We divided by 2 four times, so $a = 4$. 9. **The remaining factor is $b = 13$, which is prime and not divisible by 2.** 10. **Final prime factorization:** $$208 = 2^4 \times 13$$ **Answer:** $a = 4$, $b = 13$