How many numbers between -11 and 11 are multiples of 2 or 3?

Options

A. 11
B. 14
C. 15
D. None of these

Answer

To find how many numbers between -11 and 11 are multiples of 2 or 3, we can follow these steps:

### Step 1: Identify the range
We need to consider the integers from -11 to 11, inclusive. This means we will look at the numbers: -11, -10, -9, ..., 0, ..., 9, 10, 11.

### Step 2: Find multiples of 2
The multiples of 2 in this range are:
- From negative side: -10, -8, -6, -4, -2, 0
- From positive side: 2, 4, 6, 8, 10

So, the multiples of 2 between -11 and 11 are:
-10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10

Counting these, we find there are **11 multiples of 2**.

### Step 3: Find multiples of 3
The multiples of 3 in this range are:
- From negative side: -9, -6, -3, 0
- From positive side: 3, 6, 9

So, the multiples of 3 between -11 and 11 are:
-9, -6, -3, 0, 3, 6, 9

Counting these, we find there are **7 multiples of 3**.

### Step 4: Find common multiples of 2 and 3 (i.e., multiples of 6)
The common multiples of 2 and 3 are the multiples of 6. In this range, the multiples of 6 are:
- From negative side: -6, 0
- From positive side: 6

So, the common multiples of 2 and 3 between -11 and 11 are:
-6, 0, 6

Counting these, we find there are **3 common multiples**.

### Step 5: Apply the principle of inclusion-exclusion
To find the total number of integers that are multiples of 2 or 3, we use the formula:
\[ \text{Total} = (\text{Multiples of 2}) + (\text{Multiples of 3}) - (\text{Common multiples}) \]

Substituting the values we found:
\[ \text{Total} = 11 + 7 - 3 \]
\[ \text{Total} = 15 \]

### Conclusion
Thus, there are **15 numbers between -11 and 11 that are multiples of 2 or 3**.

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