To determine how many numbers between $1$ and $100$ (inclusive) are divisible by $3$ or $2$, we will use the principle of inclusion and exclusion.

Step 1: Find numbers divisible by $3$

We first need to calculate how many numbers between $1$ and $100$ are divisible by $3$. The numbers divisible by $3$ form the arithmetic sequence:

$3, 6, 9, 12, \dots$

To find the largest multiple of $3$ less than or equal to $100$, we divide $100$ by $3$ and round down:

$\left\lfloor \frac{100}{3} \right\rfloor = 33$

Thus, the numbers divisible by $3$ are $3 \times 1, 3 \times 2, \dots, 3 \times 33$, so there are $33$ numbers divisible by $3$.

Step 2: Find numbers divisible by $2$

Next, we calculate how many numbers between $1$ and $100$ are divisible by $2$. These numbers form the arithmetic sequence:

$2, 4, 6, 8, \dots$

To find the largest multiple of $2$ less than or equal to $100$, we divide $100$ by $2$ and round down:

$\left\lfloor \frac{100}{2} \right\rfloor = 50$

Thus, the numbers divisible by $2$ are $2 \times 1, 2 \times 2, \dots, 2 \times 50$, so there are $50$ numbers divisible by $2$.

Step 3: Find numbers divisible by both $3$ and $2$

Now, we need to find how many numbers are divisible by both $3$ and $2$. These numbers are divisible by the least common multiple (LCM) of $3$ and $2$, which is $6$. The numbers divisible by $6$ form the arithmetic sequence:

$6, 12, 18, 24, \dots$

To find the largest multiple of $6$ less than or equal to $100$, we divide $100$ by $6$ and round down:

$\left\lfloor \frac{100}{6} \right\rfloor = 16$

Thus, the numbers divisible by $6$ are $6 \times 1, 6 \times 2, \dots, 6 \times 16$, so there are $16$ numbers divisible by both $3$ and $2$.

Step 4: Apply the principle of inclusion and exclusion

To find the total number of numbers divisible by $3$ or $2$, we use the principle of inclusion and exclusion:

$\text{Total} = (\text{Divisible by 3}) + (\text{Divisible by 2}) - (\text{Divisible by both 3 and 2})$

Substituting the values we calculated:

$\text{Total} = 33 + 50 - 16 = 67$

Final Answer:

There are $67$ numbers between $1$ and $100$ that are divisible by $3$ or $2$.

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Would you like a more detailed explanation of any of the steps or have other questions on this topic? Here are some related questions you might find interesting:

1. How many numbers between $1$ and $100$ are divisible by $5$?
2. What is the general principle of inclusion and exclusion?
3. How would the answer change if we were asked for numbers divisible by $4$ or $5$?
4. What is the least common multiple (LCM) of two numbers and how is it calculated?
5. How can we apply this technique to find numbers divisible by any combination of integers?

Tip: Always remember that inclusion and exclusion is a powerful technique when dealing with overlaps between sets!