Question

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5$.

Answer 1

Step $1$: Finding sum of integers from $1$ to $100$ divisible by $2$.

Integers divisible by $2$ between $1$ to $100$ are

$2,4,6,8,\dots 100$

This is an A.P. as

First term = $a=2$

common difference $d=4-2=2$

Last term = $l=100$

$2,4,6,8,\dots .100$ Number of terms $=n=\frac{100}{2}=50$

$S_{50}=\frac{50}{2}[2+100]$

$\Rightarrow$ $S_{50}=\frac{50}{2}\left[102\right]$

$S_{50}=\frac{50}{2}\left[102\right]$

$S_{5}0=50\times 51$

$\Rightarrow S_{50}=2550$ $...\left(i\right)$

Hence, the sum of integers from $1$ to $100$ divisible by $2$ is $2550$.

$S_{50}=2550...\left(i\right)$

Step $2$. Finding sum of integers from $1$ to $100$ divisible by $5$.

Integers divisible by $5$ between $1$ to $100$ are

$5,10,15,20,\dots 100$

This is an A.P. as

First term $=a=5$

common difference $d=10-5=5$

Last term $=a_n=l=100$

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

$\Rightarrow 100=5+(n-1)5$

$\Rightarrow 100-5=(n-1)5$

$\Rightarrow 95=(n-1)5$

$\Rightarrow$ $\frac{95}{5}=(n-1)$

$\Rightarrow n=19+1$

$\Rightarrow n=20$

So,

$S_{20}=\frac{20}{2}[5+100]$

$S_{20}=10\times 105$

$\Rightarrow$ $S_{20}=1050$ $...\left(ii\right)$

Hence, the sum of integers from $1$ to $100$ divisible by $5$ is $1050$.

$S_{50}=2550...\left(i\right)$

$S_{20}=1050...\left(ii\right)$

Step $3$. Finding sum of integers from $1$ to $100$ divisible by both $2$ and $5$.

Integers divisible by $2$ and $5$ between $1$ to $100$ are,

$10,20,30,\dots \dots \dots 90,100$

This is an A.P. as

First term $=a=10$

common difference $d=20-10=10$

Last term $=a_n=l=100$

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

$\Rightarrow 100=10+(n-1)10$

$\Rightarrow 100-10=(n-1)10$

$\Rightarrow 90=(n-1)10$

$\Rightarrow$ $\frac{90}{10}=(n-1)$

$\frac{90}{10}=(n-1)$

$\Rightarrow n=9+1$

$\Rightarrow n=10$

So,

$S_{10}=\frac{10}{2}[10+100]$

$\Rightarrow S_{10}=5\times 110$

$S_{10}=550$ $...\left(iii\right)$

Hence, the sum of integers from $1$ to $100$ divisible by $2$ and $5$ is $550$.

$S_{50}=2550$ $...\left(i\right)$

$S_{20}=1050$ $...\left(ii\right)$

$S_{10}=550$ $...\left(iii\right)$

Step $4$. Finding sum of integers from $1$ to $100$ divisible by $2$ or $5$

Sum of integers divisible by $2$ or $5=$ (sum of integers divisible by $2$) $+$ ( sum of integers divisible by $5$) $-$ (sum of integers divisible by $2\&5$)

Putting values from $\left(i\right),\left(ii\right)$ & $\left(iii\right)$

Sum of integers divisible by $2$ or $5$= $2550+1050-550$

Sum of integers divisible by $2$ or $5=2550+1050-550$

Sum of integers divisible by $2$ or $5=3050$