Let's tackle the problem of finding how many integers from 1 to 56 are multiples of either 3 or 4, but not both.
First, we'll identify the multiples of 3 and the multiples of 4, and then we'll account for the integers that are multiples of both 3 and 4.
### Step-by-Step Solution:
1. Count multiples of 3:
- To find how many integers from 1 to 56 are multiples of 3, we divide 56 by 3:
[tex]\[
\text{Number of multiples of 3} = \left\lfloor \frac{56}{3} \right\rfloor = 18
\][/tex]
So, there are 18 multiples of 3 between 1 and 56.
2. Count multiples of 4:
- To find how many integers from 1 to 56 are multiples of 4, we divide 56 by 4:
[tex]\[
\text{Number of multiples of 4} = \left\lfloor \frac{56}{4} \right\rfloor = 14
\][/tex]
So, there are 14 multiples of 4 between 1 and 56.
3. Count multiples of both 3 and 4 (multiples of 12):
- First, we calculate the least common multiple (LCM) of 3 and 4, which is 12.
- To find how many integers from 1 to 56 are multiples of 12, we divide 56 by 12:
[tex]\[
\text{Number of multiples of 12} = \left\lfloor \frac{56}{12} \right\rfloor = 4
\][/tex]
So, there are 4 multiples of 12 between 1 and 56.
4. Calculate multiples of 3 or 4 but not both:
- We'll now calculate the number of integers that are multiples of either 3 or 4, but not both. This can be done using the principle of inclusion-exclusion:
[tex]\[
\text{Total multiples of 3 or 4} = (\text{Number of multiples of 3}) + (\text{Number of multiples of 4}) - (\text{Number of multiples of both 3 and 4})
\][/tex]
- We need multiples of 3 or 4 but not both:
[tex]\[
\text{Multiples of 3 or 4 but not both} = (\text{Number of multiples of 3}) + (\text{Number of multiples of 4}) - 2 \times (\text{Number of multiples of both 3 and 4})
\][/tex]
Substituting the numbers we calculated:
[tex]\[
\text{Multiples of 3 or 4 but not both} = 18 + 14 - 2 \times 4 = 32 - 8 = 24
\][/tex]
Thus, the number of integers from 1 to 56 that are multiples of 3 or 4 but not both is 24.
