At the gym, Hilary swims every six days runs every four days and cycles every 16 days if she did all three activities today and how many days would she do all three activity again on the same day

To solve this problem, we need to find out when Hilary will do all three activities on the same day again. This is determined by finding the **least common multiple (LCM)** of the three intervals: swimming every 6 days, running every 4 days, and cycling every 16 days.

### Step-by-step process:

1. **Prime factorization** of each interval:
   - 6 = 2 × 3
   - 4 = 2²
   - 16 = 2⁴

2. **LCM** is found by taking the highest power of each prime number appearing in the factorizations:
   - The highest power of 2 is 2⁴.
   - The highest power of 3 is 3¹.

3. Now, calculate the LCM:
   \[
   \text{LCM}(6, 4, 16) = 2^4 \times 3^1 = 16 \times 3 = 48
   \]

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

Would you like more details on this? Here are 5 related questions you might find useful:

1. How do we find the greatest common divisor (GCD) and why is it important in LCM calculations?
2. What if Hilary added a fourth activity that she does every 10 days, how would that affect the LCM?
3. Can the method of prime factorization be used to find LCM of larger numbers?
4. Why do we take the highest powers of prime factors for LCM?
5. What are some real-life applications of LCM in scheduling or planning?

**Tip:** The LCM helps synchronize events that occur in cycles, making it useful in planning periodic tasks or events like exercise routines or maintenance schedules.

At the gym, Hilary swims every six days, runs every four 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?

Find out how to determine when Hilary will swim, run, and cycle on the same day again using the least common multiple (LCM) of 6, 4, and 16. This math problem focuses on LCM and prime factorization. Perfect for students in Grades 5-8.

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