Divisibility Count: Integers 1 to 100 by 3 or 4

This problem requires finding the number of integers between 1 and 100 (inclusive) that are divisible by 3 or 4. We can use the Principle of Inclusion-Exclusion.

Applying Inclusion-Exclusion Principle

The principle states that for two sets A and B, the size of their union is: $|A \cup B| = |A| + |B| - |A \cap B|$

In this context:

• Set A: Integers from 1 to 100 divisible by 3.
• Set B: Integers from 1 to 100 divisible by 4.
• $A \cap B$: Integers from 1 to 100 divisible by both 3 and 4 (i.e., divisible by 12).

Calculating Set Sizes

1. Integers divisible by 3: The number of integers is calculated by $\lfloor \frac{100}{3} \rfloor$. $|A| = \lfloor \frac{100}{3} \rfloor = 33$.
2. Integers divisible by 4: The number of integers is calculated by $\lfloor \frac{100}{4} \rfloor$. $|B| = \lfloor \frac{100}{4} \rfloor = 25$.
3. Integers divisible by 12 (both 3 and 4): The least common multiple (LCM) of 3 and 4 is 12. The number of integers is calculated by $\lfloor \frac{100}{12} \rfloor$. $|A \cap B| = \lfloor \frac{100}{12} \rfloor = 8$.

Final Calculation

Now, apply the inclusion-exclusion formula:

Number divisible by 3 or 4 = (Number divisible by 3) + (Number divisible by 4) - (Number divisible by 12)

Count = $|A| + |B| - |A \cap B|$

Count = $33 + 25 - 8$

Count = $58 - 8$

Count = $50$.

Therefore, there are 50 positive integers between 1 and 100 that are divisible by 3 or 4.