Find the number of positive integers not exceeding 108 that are not divisible by 5 or by 7.

75

Step-by-step explanation:

The set of positive integer not exceeding 108 divisible by 5 is

and the set of positive integer not exceeding 108 divisible by 7 is

Moreover, there are exactly three positive numbers not exceedng 108 that are divisible by both 5 and 7, i.e,

.

Also note that the size of is , the size of is and .

On the other hand, If a positive integer not exceding 108 is not divisible by 5 or 7, then it doesn't belong to any of this sets. Therefore, the number of positive interges not exceding 108 that are not divisible by 5 or 7 is equal to

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: