Question

Let S be the set of integers x such that
I. 100 < x < 200,
II. x is odd
III. x is divisible by 3 but not by 7?
How many elements does S contain?

• A. 16
• B. 12
• C. 11
• D. 13

Answer 1

The correct option is D 13

no' of odd no. divisible by 3 not by 7 & b/w

$100<x<200$ are $\therefore$

no' of elements divisible by $[3-(by6)-(by21)+(by42\left)\right]$

$\bullet$ no' of elements divisible by '3' & b/w 100 to 200

is $\left[\frac{200}{3}\right]-\left[\frac{100}{3}\right]=66-33=33$

$\bullet$ no' of elements divisible by '6' & b/w 100 to 200

is $\left[\frac{200}{6}\right]-\left[\frac{100}{6}\right]=33-16=17$

$\bullet$ no' of elements divisible by '21' & b/w 100 to 200 is

$\left[\frac{200}{21}\right]-\left[\frac{100}{21}\right]=9-4=5$

$\bullet$ no' of elements divisible by '42' & b/w 100 to 200

$\left[\frac{200}{42}\right]-\left[\frac{100}{42}\right]=4-2=2$

$\{where\left[\right]istheq.i.f\}$

$\Rightarrow$ no' of elements $=33-17-5+2=13$