+1 vote
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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. $2916$

edited | 149 views
0
The set given in the question ends at 500, whereas answer considers it 5000. Please get the details in question corrected.

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

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.

by Loyal (5.3k points)
edited by
0
"3+3*(n-1)=4998"

Can you explain this equation a bit more...
+1

@chirudeepnamini

formula of $n^{th}$ term of Arithmetic Progression is used here to count the number of terms in a set.

+1
Now i can understand it.thanks for quick reply..

I have one more doubt

How did you calculate the highest number less than 5000 which is divisible by 12??i mean is there any method you followed?
+1

@chirudeepnamini

I don't know any other method. I have started with 5000 and take numbers less than it and check it until we get a number which is divisible by 12...It will not take much time because in worst case, I have to check 12 numbers for divisibility by 12 because remainder can be in {0,1,...11} in sequence of numbers..or check whether a number is divisible by 3 and 4 both...

+1
Got it now thanks for the explanation:)