Consider a set of integers from 1 to 250.

(A) Find how many ways of these numbers are divisible by 3 or 5 or 7? Also

(B) Indicate how many are divisible are divisible by 3 or 5 but not by7?

Solution:

$\rightarrow \quad x=250$

$\quad|A|=\left|\frac{250}{3}\right|=83$

$\quad|B|=\left|\frac{250}{5}\right|=50$

$\quad|C|=\left|\frac{250}{7}\right|=35$

$ \quad|A \cap B|=\left|\frac{250}{3 \times 5}\right|=16$

$ \quad|B \cap C|=\left|\frac{250}{5 \times 7}\right|=7$

$ \quad|A \cap c|=\left|\frac{250}{3 \times 7}\right|=11$

$\quad|A \cap B \cap C|=\left|\frac{250}{3 \times 5 \times 7}\right|=2$

(A) Find how many ways of these numbers are divisible by 3 or 5 or 7:

$\rightarrow|A \cup B \cup C|=|A|+|B|+|C|-|A \cap B|-|B \cap C|-|A \cap C|+|A \cap B \cap C|$

$=83+50+35-16-7-11+2$

$=136$

(B) Indicate how many are divisible are divisible by 3 or 5 but not by7:

$\rightarrow|A \cup B|=|A|+|B|-|A \cap B|$

$=83+50-16$

$=177 .$