Question

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

• A. $752500$
• B. $75250$
• C. $7525$
• D. $70250$

Answer 1

Answer: (B)

$75250$

$N1$: 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 $500=$
$S_{N1}=\frac{N1}{2}[2a+(N1-1\left)d\right]$
$S_{N1}=\frac{250}{2}\left[2\right(2)+(250-1\left)2\right]S_{N1}=125\left[502\right]=62750$
Sum of integers which are multiples of $5$ from $1$ to $500=$
$S_{N2}=\frac{100}{2}\left[2\right(5)+(100-1\left)5\right]S_{N2}=50\left[505\right]=25250$
Sum of integers which are multiples of $10$ from $1$ to $500=$
$S_{N3}=\frac{50}{2}\left[2\right(10)+(50-1\left)10\right]S_{N3}=25\left[510\right]=12750$
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 $=75250$