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

To find out when Hilary will perform all three activities (swimming, running, and cycling) on the same day again, we need to find the Least Common Multiple (LCM) of the three numbers: 6 (days for swimming), 4 (days for running), and 16 (days for cycling).

### Steps:
1. **Prime factorization** of each number:
   - \( 6 = 2 \times 3 \)
   - \( 4 = 2^2 \)
   - \( 16 = 2^4 \)

2. **LCM** is found by taking the highest power of each prime factor:
   - The highest power of 2 is \( 2^4 \).
   - The highest power of 3 is \( 3^1 \).

3. Multiply the highest powers of each prime factor to get the LCM:
   \[
   \text{LCM} = 2^4 \times 3 = 16 \times 3 = 48
   \]

Thus, Hilary will perform all three activities again on the same day in **48 days**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. What is the general method for finding the LCM of multiple numbers?
2. How is the prime factorization of a number useful in determining the LCM?
3. If Hilary swims every 6 days but only cycles every 32 days,

At the gym Hilary 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 the three activities again on the same day?

Learn how to calculate when Hilary will swim, run, and cycle on the same day again by finding the Least Common Multiple (LCM) of 6, 4, and 16. This problem covers key concepts like LCM and prime factorization, making it suitable for students in Grades 6-8.

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