Question

19. Find the sum of all the natural numbers less than $1000$ and which are neither divisible by $5$ nor by $2$.

Answer 1

Step 1. Note the given data:

Given: A series of natural numbers less than $1000$ which is neither divisible by $5$ nor by $2$.

Natural number: A set of non-negative numbers excluding $0$. i.e. $1,2,3,4,.......n$

An odd number is a number that is not divisible by $2$.

Series of odd numbers less than $1000$ is $1,3,5,7,.......999$

The first term of the odd number $a=1$

Step 2. Find the common difference.

The formula for the common difference of an AP is $d=t_n-t_{n-1}$

consider $n=2$

Therefore,$t_n=a_2=3,t_{n-1}=t_{2-1}=t_1=1$

Substituting the value.

$$\begin{gathered} d=3-1 \\ d=2 \end{gathered}$$

Step 3. Find the value of number of terms

The formula of $n^{th}$term of an AP is$t_n=a+\left(n-1\right)d$

Here $t_n=999,a=1,d=2$

Substituting the value

$$\begin{gathered} 999=1+(n-1)2 \\ 999=1+2n-2 \\ 999=2n-1 \end{gathered}$$

Add $1$ on both sides

$$\begin{gathered} 999+1=2n-1+1 \\ 1000=2n \end{gathered}$$

Divide by $2$ on both sides

$$\begin{gathered} \frac{2n}{2}=\frac{1000}{2} \\ n=500 \end{gathered}$$

Hence the number of terms of odd numbers less than $1000$ is $500$.

Step 4. Find the sum of the $500$ terms of first odd numbers.

The formula of the sum of the first $n$ term of an AP is$S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

Here $a=1,d=2$

Substituting the value

$$\begin{gathered} S_{500}=\raisebox{1ex}{$500$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(2\left(1\right)+\left(500-1\right)2\right) \\ {S}_{500}=\raisebox{1ex}{$500$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(2+\left(499\right)2\right) \\ {S}_{500}=250\left(2+998\right) \\ {S}_{500}=250\times 1000 \\ {S}_{500}=250000 \end{gathered}$$

Step 5. Find the series of odd numbers which are divisible by $5$.

Series of odd numbers which are divisible by $5$ less than$1000$ is $5,15,25,.......995$

The first term of the odd number which is divisible by $5$ is $a=5$

Step 6 Find the common difference.

The formula for the common difference of an AP is $d=t_n-t_{n-1}$

consider $n=2$

Therefore,$t_n=t_2=15,t_{n-1}=t_{2-1}=t_1=5$

Substituting the value.

$$\begin{gathered} d=15-5 \\ d=10 \end{gathered}$$

Step 7. Find the value of the number of terms

The formula of $n^{th}$term of an AP is$a_n=a+\left(n-1\right)d$

Here $t_n=995,a=5,d=10$

Substituting the value

$$\begin{gathered} 995=5+(n-1)10 \\ 995=5+10n-10 \\ 995=10n-5 \end{gathered}$$

Add $5$ on both sides

$$\begin{gathered} 995+5=10n-5+5 \\ 1000=10n \end{gathered}$$

Divide by $10$ on both sides

$$\begin{gathered} \frac{10n}{10}=\frac{1000}{10} \\ n=100 \end{gathered}$$

Hence the number of terms of odd numbers which are divisible by $5$ but less than $1000$ is $100$.

Step 8. Find the sum of the first $100$ odd numbers which are divisible by $5$ .

The formula of sum of the $n^{th}$ term of an AP is $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$

Here $a=5,d=10$

Substituting the value

$$\begin{gathered} S_{100}=\frac{100}{2}\left[2\left(5\right)+\left(100-1\right)10\right] \\ {S}_{100}=\frac{100}{2}\left[10+\left(99\right)10\right] \\ {S}_{100}=50\left(10+990\right) \\ {S}_{100}=50\left(1000\right) \\ {S}_{100}=50000 \end{gathered}$$

Step 9. Find the sum of all the natural numbers less than $1000$ which are neither divisible by $2$ nor by $5$.

The sum is the difference between the sum of odd numbers less than$1000$ and sum of odd numbers which are divisible by $5$ less than $1000$.

$$\begin{gathered} S=S_{500}-S_{100} \\ S=250000-50000 \\ S=200000 \end{gathered}$$

Hence, the sum of all the natural numbers less than $1000$ which are neither divisible by $2$ nor by $5$ is $200000$.