Question

How many positive integers less than 100 is not a factor of 2,3 and 5?

Answer 1

Let $A,B$ and $C$ denote the sets of positive integers that are divisible by 2, 3 and 5 respectively and less or equal to 100.

Then $|A|=\lfloor \frac{100}2\rfloor=50, |B|=\lfloor \frac{100}3\rfloor=33,$ and $|C|=\lfloor \frac{100}5\rfloor=20.$

If $m$ and $n$ are relatively prime then a $a$ is divisible by $m$ and $n$ iff it is divisible by $mn.$

It follows that $|A\cap B|=\lfloor \frac{100}6\rfloor=16, |A\cap C|=\lfloor \frac{100}{10}\rfloor=10, |B\cap C|=\lfloor \frac{100}{15}\rfloor=6,$ and $|A\cap B\cap C|=\lfloor \frac{100}{30}\rfloor=3.$

Therefore, by Inclusion-exclusion principle, the number of positive integers that are divisible by 2 or 3 or 5 and less or equal to 100 is equal to

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C| \\=50+33+20-16-10-6+3=74.$

We conclude that the number of integers that are not divisible by 2 or 3 or 5 is equal to $100-74=26.$