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GATE IT 2006 | Question: 24

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34 34 votes

What is the cardinality of the set of integers $X$ defined below?

$X=\{n \mid 1 \leq n ≤ 123, n$ is not divisible by either $2$, $3$ or $5\}$

  1. $28$
  2. $33$
  3. $37$
  4. $44$

3 Answers

Best answer
71 71 votes

Number’s divisible by $2$ in $X = 61$        [ = integer(123/2) ]

Number’s divisible by $3$ in $X = 41$ 

Number’s divisible by $5$ in $X = 24$

Number’s divisible by $2$ and $3$ i.e. by $6 = 20$ 

Number’s divisible by $2$ and $5$ i.e by $10 = 12$

Number’s divisible by $3$ and $5$ i.e by $15 = 8$ 

Number’s divisible by $2$ and $3$ and $5$ i.e by $30 = 4$ 

Number’s divisible by either $2$ or $3$ or $5$ = $N(AUBUC)$ = $N(A) +N(B)+N(C) -N(A∩B)-N(B∩C)-N(A∩C)+ N(A∩B∩C) $

$= 61 +41+24 -20-12-8 +4 = 90$ 

$X$={ $n ,1 ≤ n ≤ 123, n$ is not divisible by either $2, 3$ or $5$ }  

Cardinality = $123-90$ =$33$

Correct Answer: $B$

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4 4 votes
Number of elements that are divisible by a and b in between 1 to n are = $\left \lfloor n/(l.c.m(a.b))\right \rfloor$.
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