Question

Eve is making 50 gift baskets for the holidays. She wraps a red bow on every three baskets, a blue bow on every six baskets, and a gold bow on every eight baskets. Which baskets have all three bows?

Answer 1

To determine which gift baskets have all three colored bows, we find the least common multiple of 3, 6, and 8, which is 24. Thus, every 24th basket will have a red, blue, and gold bow, specifically baskets number 24 and 48.

Step-by-step explanation:

The question asks us to determine which gift baskets have a red, blue, and gold bow, given that a red bow is placed on every third basket, a blue bow on every sixth basket, and a gold bow on every eighth basket. To solve this, we need to find the least common multiple (LCM) of the numbers 3, 6, and 8, which is the smallest number that all three numbers can divide into without leaving a remainder.

To find the LCM of 3, 6, and 8, we can prime factorize these numbers:

• 3 is a prime number, so its prime factorization is just 3.
• 6 is 2 × 3.
• 8 is 2 × 2 × 2.

Now, we take the highest powers of all the prime factors that appear: 2³ (from 8) and 3¹ (from 3 and 6). Multiplying these together gives us 2³ × 3¹ = 8 × 3

= 24.

The LCM of 3, 6, and 8 is 24, meaning that every 24th basket will have a red, blue, and gold bow. Since Eve is making 50 baskets, the baskets with all three bows will be basket numbers 24 and 48.