To solve the problem of finding how many multiples of 2, 3, or 7 are less than or equal to 210, we can use the principle of inclusion-exclusion.

Steps:

1. Find the multiples of 2, 3, and 7 up to 210:

   • Multiples of 2 up to 210: $\left\lfloor \frac{210}{2} \right\rfloor = 105$
   • Multiples of 3 up to 210: $\left\lfloor \frac{210}{3} \right\rfloor = 70$
   • Multiples of 7 up to 210: $\left\lfloor \frac{210}{7} \right\rfloor = 30$

2. Find the multiples of intersections (LCMs of pairs):

   • Multiples of both 2 and 3 (LCM of 2 and 3 = 6): $\left\lfloor \frac{210}{6} \right\rfloor = 35$
   • Multiples of both 2 and 7 (LCM of 2 and 7 = 14): $\left\lfloor \frac{210}{14} \right\rfloor = 15$
   • Multiples of both 3 and 7 (LCM of 3 and 7 = 21): $\left\lfloor \frac{210}{21} \right\rfloor = 10$

3. Find the multiples of 2, 3, and 7 (LCM of 2, 3, and 7 = 42):

   • Multiples of 2, 3, and 7 (LCM 42): $\left\lfloor \frac{210}{42} \right\rfloor = 5$

4. Apply inclusion-exclusion principle:

$\text{Total} = (105 + 70 + 30) - (35 + 15 + 10) + 5$ $\text{Total} = 205 - 60 + 5 = 150$

Thus, there are 150 multiples of 2, 3, or 7 less than or equal to 210.

Would you like further clarification or details on this method?

Here are 5 related questions to deepen your understanding:

1. How would you calculate multiples of any other set of numbers using inclusion-exclusion?
2. What is the importance of least common multiples (LCMs) in problems involving multiples?
3. How does the principle of inclusion-exclusion work in more than three sets?
4. How can you generalize this solution for multiples less than any other number (e.g., 500)?
5. What happens if the numbers in the problem are not pairwise coprime (e.g., multiples of 2, 4, and 6)?

Tip: When applying the inclusion-exclusion principle, always make sure to subtract overlaps (intersections of two sets) and add back overlaps of three sets to avoid over-counting.