Find the sum of those integers from $1$ to $500$ which are multiples of  $2$ or $5.$

$N1$: Number of terms from $1$ to $500$ which are multiples of $2$ $=250$
$N1:2,4,6,\mathrm{8............500}$
$N2$: Number of terms from $1$ to $500$ which are multiples of $5$ $=100$
$N2:5,10,\mathrm{15.......500}$
$N3$: Number of terms from $1$ to $500$ which are multiples of $10$ (both $2$ and $5$) $=50$
$N3:10,20,\mathrm{30...500}$
Sum of integers which are multiples of $2$ from $1$ to $500=$ 
$S_{N1}=\frac{N1}{2}[2a+(N1-1)d]$
$$\begin{gathered} S_{N1}=\frac{250}{2}[2(2)+(250-1)2] \\ {S}_{N1}=125[502]=62750 \end{gathered}$$
Sum of integers which are multiples of $5$ from $1$ to $500=$
$$\begin{gathered} S_{N2}=\frac{100}{2}[2(5)+(100-1)5] \\ {S}_{N2}=50[505]=25250 \end{gathered}$$
Sum of integers which are multiples of $10$ from $1$ to $500=$
$$\begin{gathered} S_{N3}=\frac{50}{2}[2(10)+(50-1)10] \\ {S}_{N3}=25[510]=12750 \end{gathered}$$
Hence, the sum of integers from $1$ to $500$ which are multiples of $2$ or $5=$ 
$S_{N1}+S_{N2}-S_{N3}$
$=62750+25250-12750$
$=75250.$
Sum $=75250$