1. **State the problem:** We need to find the shortest distance from their home to the theater such that both Mark and Jodi use a whole number of gallons of gas. 2. **Given:** - Mark's car mileage = 30 miles per gallon - Jodi's car mileage = 21 miles per gallon 3. **Key idea:** The distance must be a multiple of both 30 and 21 so that when divided by their respective mileages, the result is an integer (whole number of gallons). 4. **Find the least common multiple (LCM) of 30 and 21:** - Prime factorization: $$30 = 2 \times 3 \times 5$$ $$21 = 3 \times 7$$ - LCM is the product of the highest powers of all prime factors: $$\text{LCM} = 2 \times 3 \times 5 \times 7 = 210$$ 5. **Interpretation:** The shortest distance that is divisible by both 30 and 21 is 210 miles. 6. **Check:** - Mark's gallons: $$\frac{210}{30} = 7$$ (whole number) - Jodi's gallons: $$\frac{210}{21} = 10$$ (whole number) **Final answer:** The shortest distance is **210 miles**.