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Consider the set of integers $\{1,2,3,\ldots,5000\}.$ The number of integers that is divisible by neither $3$ nor $4$ is $:$

1. $1668$
2. $2084$
3. $2500$
4. $2916$

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The set given in the question ends at 500, whereas answer considers it 5000. Please get the details in question corrected.

+1 vote

$S = \{1,2,3,\ldots,4999,5000\}$

Let 'A' be the set of integers divisible by $3 \implies A =\{3,6,9,12,15,\ldots,4998\}$

Let 'B' be the set of integers divisible by $4 \implies B= \{4,8,12,16,\ldots,5000\}$

Let 'C' be the set of integers divisible by both $3$ and $4 \implies C = \{12,24,36,\ldots,4992\}$

$|A| = 3+(n-1)3 = 4998 \implies n = 4998/3 = 1666$

$|B| = 4+(n-1)4 = 5000 \implies n = 5000/4 = 1250$

$|C| =12+(n-1)12 = 4992 \implies n = 4992/12 = 416$

Divisible by $3$ or $4:$  $\{3,4,6,8\dots 5000\}$ : $|D| = |A| + |B| -|C|$

$= 1666+1250-416$

$= 2500$

Number of integers that is divisible by neither $3$ nor $4$

$= |S| - |D|$

$= 5000-2500$

$= 2500$

So, C is the correct answer.

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