1. **State the problem:** Sharon has a blood test every 75 days and a fitness test every 110 days. Both tests were done on the same day before the training started. We need to find after how many days both tests will occur again on the same day. 2. **Understand the problem:** The problem asks for the next time both events coincide. This is the least common multiple (LCM) of 75 and 110. 3. **Find the prime factors:** - 75 = $3 \times 5^2$ - 110 = $2 \times 5 \times 11$ 4. **Calculate the LCM:** The LCM is the product of the highest powers of all prime factors: $$\text{LCM} = 2 \times 3 \times 5^2 \times 11 = 2 \times 3 \times 25 \times 11$$ 5. **Multiply step-by-step:** - $2 \times 3 = 6$ - $6 \times 25 = 150$ - $150 \times 11 = 1650$ 6. **Conclusion:** Sharon will next have both a blood test and a fitness test on the same day after **1650 days**.