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The question asks how many days it will take for Hillary to perform all three activities—swimming, running, and cycling—on the same day again. The key to solving this problem is to find the least common multiple (LCM) of the three activity cycles: 6 days (swimming), 4 days (running), and 16 days (cycling).

We calculate the LCM of 6, 4, and 16 by determining the prime factorizations of each:

- 6 = \(2 \times 3\)
- 4 = \(2^2\)
- 16 = \(2^4\)

The LCM is the product of the highest powers of all prime factors:
- The highest power of 2 is \(2^4\)
- The highest power of 3 is \(3\)

Therefore, the LCM is:
\[
LCM(6, 4, 16) = 2^4 \times 3 = 16 \times 3 = 48
\]

So, Hillary will do all three activities again on the same day in 48 days.

Would you like further details or have any questions?

Here are five related questions you may find helpful:
1. What is the general method for finding the least common multiple of multiple numbers?
2. How do prime factorizations help in calculating the LCM?
3. Can the greatest common divisor (GCD) also be useful in scheduling problems?
4. How would the answer change if one of the activity cycles were different, e.g., every 5 days instead of 6?
5. What other real-life situations can be modeled by finding the LCM?

**Tip:** Understanding prime factorization can be useful not only for LCM but also for simplifying fractions and solving Diophantine equations.

At the gym, Hillary swims every 6 days, runs every 4 days, and cycles every 16 days. If she did all three activities today, in how many days will she do all three activities again on the same day?

In this problem, Hillary swims every 6 days, runs every 4 days, and cycles every 16 days. Learn how to calculate when she will perform all three activities on the same day using the Least Common Multiple (LCM) method. This problem is suitable for students in Grades 6-8 and covers LCM concepts and prime factorization.

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