Question

The sum of the integers from $1$ to $100$ which are not divisible by $3$ or $5$ is

• A. $2489$
• B. $4735$
• C. $2317$
• D. $2632$

Answer 1

The correct option is D

$2632$

Explanation for the correct option:

Step 1: Calculate the sum of the first $100$ natural numbers

Sum of first $n$ natural numbers $=\frac{n\left(n+1\right)}{2}$

Sum of first $100$ natural numbers $=\frac{100\left(100+1\right)}{2}$
$$\begin{gathered} =\frac{100\left(101\right)}{2} \\ =5050 \end{gathered}$$

Step 2: Calculate the sum of multiples of $3$ upto $100$

Integers divisible by $3$ upto $100$ are $3,6,9,\dots ,99$

These are in arithmetic progression (AP).

General form of an AP is
$a_n=a+\left(n-1\right)d$
Where, $a$ is the first term, $d$ is common difference, and $a_n$ is the $n$^th term.

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

Sum of an AP is given as
$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

Thus, the sum of the multiples of $3$ is

$\begin{array}{rcl}{S}_{33}& =& \frac{33}{2}\left[2\times 3+\left(33-1\right)3\right]\\ & =& 1683\end{array}$

Step 3: Calculate the sum of multiples of $5$ upto $100$

Integers divisible by $5$ upto $100$ are $5,10,15,\dots ,100$

These are in arithmetic progression (AP). Thus, from the general formula for $n$^th term of an AP,

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

Thus, the sum of the multiples of $3$ is

$\begin{array}{rcl}{S}_{20}& =& \frac{20}{2}\left[2\times 5+\left(22-1\right)5\right]\\ & =& 1050\end{array}$

Step 4: Find the sum of multiples of $3$ and $5$ upto $100$

Integers divisible by $15$ upto $100$ are $15,30,45\dots ,90$

These are in arithmetic progression (AP). Their common difference is $15$.
Thus, from the general formula for $n$^th term of an AP,

$$\begin{gathered} \Rightarrow 90=15+(n-1)15 \\ \Rightarrow n=6 \end{gathered}$$

The sum of the multiple of $3$ and $5$ is

$\begin{array}{rcl}{S}_6& =& \frac{6}{2}\left[2\times 15+\left(6-1\right)15\right]\\ & =& 315\end{array}$

Step 5: Find the required sum

So, the sum of integers from $1$ to $100$ which are not divisible by $3$ or $5$ $=$The sum of the first $100$ natural numbers
$-$the sum of multiples of $3$$-$the sum of multiples of $5$
$+$ the sum of the multiples of both $3$ and $5$

$=5050-1683-1050+315=2632$

(Since the terms divisible by $3$ and $5$ are accounted for in their respective multiples, they get subtracted twice. Thus, we need to add them once)

Hence, the correct option is D.