Question

How many integers between $1$ and $10,000$, inclusive, are not divisible by $4$, not divisible by $5$, and not divisible by $6$?

Clearly show your work.

Answer 1

Step-1: Find the numbers divisible by $4$,$5$ and $6$ between $1$ and $10,000$.

Let $A,B,C$ and $D$ be the sets of integers between $1$ and $10,000$(inclusive) which are divisible by $4$, by $5$, and by $6$ respectively.

Let the number divisible by $4$ between $1$and $10,000$ be $\mathrm{n}\left(\mathrm{A}\right)$.

$$\begin{gathered} \mathrm{n}\left(\mathrm{A}\right)=\frac{10,000}{4} \\ =2,500 \end{gathered}$$

Let the number divisible by $5$ between $1$and $10,000$ be $\mathrm{n}\left(\mathrm{B}\right)$

$$\begin{gathered} \mathrm{n}\left(\mathrm{B}\right)=\frac{10,000}{5} \\ =2000 \end{gathered}$$

Let the number divisible by $6$ between $1$and $10,000$ be $n\left(C\right)$.

$$\begin{gathered} \mathrm{n}\left(\mathrm{C}\right)=\frac{10,000}{6} \\ =1666 \end{gathered}$$

Step-2: Find the numbers which are divisible by $4$ and $5$, $5$ and $6$, $4$ and $6$ between $1\mathrm{to}10,000$.

Numbers that are divisible by $4,5$and $6$ between $1\mathrm{to}10,000$.

That is $\mathrm{n}(\mathrm{A}\cup \mathrm{B}\cup \mathrm{C})$.

Let the number divisible by $4,5$ be $\mathrm{n}(\mathrm{A}\cap \mathrm{B})$.

$$\begin{gathered} \mathrm{n}(\mathrm{A}\cap \mathrm{B})=\frac{10,000}{4\times 5} \\ =500 \end{gathered}$$

Let the number divisible by $5$ and $6$ be $\mathrm{n}(\mathrm{B}\cap \mathrm{C})$.

$$\begin{gathered} \mathrm{n}(\mathrm{B}\cap \mathrm{C})=\frac{10,000}{5\times 6} \\ \approx 333 \end{gathered}$$

Let the number divisible by $4$ and $6$ be $\mathrm{n}(\mathrm{A}\cap \mathrm{C})$.

$$\begin{gathered} \mathrm{n}(\mathrm{A}\cap \mathrm{C})=\frac{10,000}{4\times 6} \\ =416 \end{gathered}$$

We take only round-off numbers here.

Step-3: Find the numbers which are divisible by $4,5$and $6$ .

Now let the numbers which are divisible by $4,5$ and $6$ be $\mathrm{n}(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C})$.

$$\begin{gathered} \mathrm{n}(\mathrm{A}\cap \mathrm{B}\cap \mathrm{C})=\frac{10,000}{4\times 5\times 6} \\ =83 \end{gathered}$$

Now substitute all the above values in:

$\mathrm{n}(\mathrm{A}\cup \mathrm{B}\cup \mathrm{C})=\mathrm{n}\left(\mathrm{A}\right)+\mathrm{n}\left(\mathrm{B}\right)+\mathrm{n}\left(\mathrm{C}\right)-\mathrm{n}(\mathrm{A}\cap \mathrm{B})-\mathrm{n}(\mathrm{B}\cap \mathrm{C})-\mathrm{n}(\mathrm{A}\cap \mathrm{C})-\mathrm{n}(\mathrm{A}\cap \mathrm{B}\cap \mathrm{B})$.

$$\begin{gathered} \mathrm{n}(\mathrm{A}\cup \mathrm{B}\cup \mathrm{C})=2,500+2000+1666-500-333-416-83 \\ =5134 \end{gathered}$$

$5134$ numbers are divisible by $4,5$and $6$.

So the numbers which are not divisible by these numbers are$10000-5134=4866$

Therefore, Integers between $1-10,000$ (inclusive) which are not divisible by $4$,$5$ and $6$ are $4866$.