Question

How many natural numbers less than 300 are neither multiples of 2 nor multiples of 3? show work please

Answer 1

Answer:100

Step-by-step explanation:

Suppose S is the set of number of first 300 natural number

$S=\{1,2,3,4.......300\}$

And A be the set of number divisible by 2

$A=\{2,4,6.......300\}$

so total element in A is $\frac{300}{2}=150$

Let B be the set containing the element divisible by 3

$B=\{3,6,9......300\}$

So element in B is $\frac{300}{3}=100$

But there are some element which is common in both A and B

so $A\cap B={6,12,18.......300}$

Element in $A\cap B=\frac{300}{6}=50$

and so number of elements which are less than 300 and neither divisible by 2  nor 3 is

$=n(S)-n(A\cup B)$

$=300-(n(A)+n(B)-n(A\cap B))$

$=300-(150+100-50)$

$=300-200$

$=100$