Question

If x denotes the value of the sum of the integers lying between 1 and 100 which are divisible by either 3 or 7. The value of X equals

• A. 5050
• B. 3000
• C. 4068
• D. 2208

Answer 1

Answer: (D)

2208

Divisible by $3:3,6,9,12,.....,99$

A.P with $a=3,d=3,a_n=99,n=?$

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

$\Rightarrow 99=3+(n-1)3$

$\Rightarrow 96=3n-3$

$\Rightarrow 3n=99$

$\Rightarrow n=33$

Sum $=\frac{n}{2}(2a+(n-1\left)d\right)$

$=\frac{n}{2}(a+a_n)=\frac{33}{2}(3+99)$

$=33\left(51\right)=1683$

Divisible by $7:7,14,\dots ..98$

$98=7+(n-1)7$

$\Rightarrow 91=7n-7$

$\Rightarrow 7n=98$

$\Rightarrow n=14$

Sum $=7(7+98)=7\times 105=735$

Divisible by both $3$ & $7$, i.e., by $21$,

$21,42,63,84$

$\Rightarrow$ Sum $=63+147=210$

$\Rightarrow$ Total sum$=1683+735-210$

$=2208$.