Question

Jackson Fitness sells gym passes in packs of 10. A competing gym, Webster Wellness, sells gym passes in sets of 4. If both sold the same number of gym passes last month, what is the smallest number of passes each could have sold

Answer 1

To find the smallest number of gym passes that both Jackson Fitness (which sells passes in packs of 10) and Webster Wellness (which sells passes in packs of 4) could have sold if they sold the same total number of passes, we need to determine the least common multiple (LCM) of the two pack sizes: 10 and 4.

1. **Factor the numbers into their prime factors:**
- The prime factorization of 10 is \( 2 \times 5 \).
- The prime factorization of 4 is \( 2^2 \).

2. **Identify the highest powers of each prime factor that appear in the factorizations:**
- For the prime factor 2, the highest power between 10 and 4 is \( 2^2 \) (from 4).
- For the prime factor 5, the highest power is \( 5^1 \) (from 10).

3. **Multiply the highest powers together to find the LCM:**
\[
\text{LCM} = 2^2 \times 5^1 = 4 \times 5 = 20.
\]

Thus, the least common multiple of 10 and 4 is 20.

4. **Conclusion:**
The smallest number of gym passes each gym could have sold last month is **20**.