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

Answer: (D)

$2632$

The sum of numbers not divisible by $3$ or $5$

= Sum of first $100$ natural numbers − Sum of the divisor of $3$− Sum of the divisor of $5$+ Sum of the divisor of $15$

Sum $={\sum }_{n=1}^{100}n-{\sum }_{k=1}^{33}3k-{\sum }_{i=1}^{20}5i-{\sum }_{j=1}^{6}15j$

$=\frac{n.n+1}{2}-\frac{3k(k+1)}{2}-\frac{5i(i+1)}{2}+\frac{15j(j+1)}{2}$

$=\frac{100\times 101}{2}-\frac{3\times 33\times 34}{2}-\frac{5\times 20\times 21}{2}+\frac{15\times 6\times 7}{2}$

$=5050-1683-1050+315$

$=2632.$

Hence, the answer is $2632.$