Question

How many numbers between $1$ and $100$ (inclusive) are divisible by $3$ or $4$?

Answer 1

Step-1: Find the number between $1$ to $100$ divisible by $3$.

The first number is $3$ and the last number is $99$.

Let the total number be $n$ and the common difference is $3$.

Recall the nth term of the arithmetic formula:

$a_n=a+(n-1)d$

Step-2: Find the number of terms.

Substitute $a=3,d=3,a_n=99$:

$$\begin{gathered} 99=3+(n-1)3 \\ \Rightarrow 99-3=(n-1)3 \\ \Rightarrow 96=(n-1)3 \\ \Rightarrow n-1=\frac{96}{3} \\ \Rightarrow n=32+1 \\ \Rightarrow n=33 \end{gathered}$$

Thus, the number of terms divisible by $3$ is $n\left(\mathrm{divisible}\mathrm{by}3\right)=33$.

Step 2: Find the number between $1$ to $100$ divisible by $4$.

Substitute $a=4,d=4,a_n=100$ in $a_n=a+(n-1)d$:

$$\begin{gathered} 99=3+(n-1)3 \\ \Rightarrow 100-4=(n-1)4 \\ \Rightarrow 96=(n-1)4 \\ \Rightarrow n-1=\frac{96}{4} \\ \Rightarrow n=24+1 \\ \Rightarrow n=25 \end{gathered}$$

Thus, the number of terms divisible by $4$is $n\left(\mathrm{divisible}\mathrm{by}4\right)=25$.

Step 2: Find the number between $1$ to $100$ divisible by $3\mathrm{and}4$.

The LCM of $3\mathrm{and}4\mathrm{is}12.$

Substitute $a=12,d=12,a_n=96$ in $a_n=a+(n-1)d$:

$$\begin{gathered} 99=3+(n-1)3 \\ \Rightarrow 96-12=(n-1)12 \\ \Rightarrow 84=(n-1)12 \\ \Rightarrow n-1=\frac{84}{12} \\ \Rightarrow n=7+1 \\ \Rightarrow n=8 \end{gathered}$$

Similarly, the number of terms divisible by $12$ is $\mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\mathrm{and}4\right)=8$.

The total number of terms between $1$ to $100$ are divisible by $3$ or $4$:

$$\begin{gathered} \mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\mathrm{or}4\right)=\mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\right)+\mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\right)+\mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\mathrm{and}4\right) \\ \Rightarrow \mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\mathrm{or}4\right)=33+25-8 \\ \Rightarrow \mathrm{n}\left(\mathrm{divisible}\mathrm{by}3\mathrm{or}4\right)=50 \end{gathered}$$

Hence, the total number between $1$ to $100$ divisible by $3\mathrm{or}4$ are $50$.