1. **State the problem:** We need to find how often a gift bag contains all three items: a hat, a T-shirt, and a backpack. According to the problem, hats are given to every 2nd visitor, T-shirts to every 7th visitor, and backpacks to every 10th visitor. 2. **Identify the mathematical concept:** To find how often a bag contains all three items, we need to find the common visitors who get all three items. This means finding the least common multiple (LCM) of the numbers 2, 7, and 10. 3. **Recall the formula for LCM:** The LCM of several numbers is the smallest positive integer that is divisible by all of them. 4. **Calculate the LCM:** - Prime factorization: - 2 = $2$ - 7 = $7$ - 10 = $2 \times 5$ - Take the highest powers of all prime factors: - $2^1$, $5^1$, $7^1$ - Multiply these together: $$\text{LCM} = 2 \times 5 \times 7 = 70$$ 5. **Interpret the result:** Every 70th visitor will receive a bag containing all three items. **Final answer:** A bag will contain all three items every $70$ visitors.