Question

The sum of integers from $1$ to $100$ that are divisible by$2$ or $5$ is

• A. $3000$
• B. $3050$
• C. $4050$
• D. None of these

Answer 1

The correct option is B

$3050$

Explanation for the correct option:

Step 1. Find the sum of the digit divisible by $2$ .

The integers which are divisible by $2$ in between $1$ and $100$ is $2,4,6,....,100$

So the first term $a_1=2$ and the common difference is $d=2$

We know that the $n^{th}$ term is given by $a_n=a_1+d\left(n-1\right)$

$a_n=100$

So $n=\frac{a_n-a_1}{d}+1$

$$\begin{gathered} \Rightarrow n=\frac{100-2}{2}+1 \\ =50 \end{gathered}$$

We know that sum of first $n$ term is $S=\frac{n}{2}\left(2a+d\left(n-1\right)\right)$

$$\begin{gathered} \therefore S=\frac{50}{2}\left(2·2+2\left(50-1\right)\right) \\ =2550 \end{gathered}$$

Step 2. Find the sum of the digit divisible by $5$ .

The integers which are divisible by $5$ in between $1$ and $100$ is $5,10,15,....,100$

So the first term $a_1=5$ and the common difference is $d=5$

We know that the $n^{th}$ term is given by $a_n=a_1+d\left(n-1\right)$

$a_n=100$

So $n=\frac{a_n-a_1}{d}+1$

$$\begin{gathered} \Rightarrow n=\frac{100-5}{5}+1 \\ =20 \end{gathered}$$

We know that sum of first $n$ term is $S=\frac{n}{2}\left(2a+d\left(n-1\right)\right)$

$$\begin{gathered} \therefore S=\frac{20}{2}\left(2·5+5\left(20-1\right)\right) \\ =1050 \end{gathered}$$

Step 3. Find the sum of the digit divisible by $2$and $5$ .

The integers which are divisible by $2$and $5$ in between $1$ and $100$ is $10,20,....,100$

So the first term $a_1=10$ and the common difference is $d=10$

We know that the $n^{th}$ term is given by $a_n=a_1+d\left(n-1\right)$

$a_n=100$

So $n=\frac{a_n-a_1}{d}+1$

$$\begin{gathered} \Rightarrow n=\frac{100-10}{10}+1 \\ =10 \end{gathered}$$

We know that sum of first $n$ term is $S=\frac{n}{2}\left(2a+d\left(n-1\right)\right)$

$$\begin{gathered} \therefore S=\frac{10}{2}\left(2·10+10\left(10-1\right)\right) \\ =550 \end{gathered}$$

Step 4. Find the sum of the digit divisible by$2$ or $5$

Required sum $=$ sum of integers divisible by $2$ $+$ sum of integers divisible by $5$ $–$ sum of integers divisible $2$ by and $5$

Required sum $$\begin{gathered} =2550+1050-550 \\ =3050 \end{gathered}$$

Therefore, the sum of integers from $1$ to $100$ that are divisible by$2$ or $5$ is $3050$.

Hence, option (B) is the correct answer.