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The probability that a given positive integer lying between $1$ and $100$ (both inclusive) is NOT divisible by $2$, $3$ or $5$ is ______ .

Answer - $0.26$

Number of integers divisible by $2 = 50$

Number of integers divisible by $3 = 33$

Number of integers divisible by $5 = 20$

Number of integers divisible by $2$ and $3 = 16$

Number of integers divisible by $2$ and $5 = 10$

Number of integers divisible by $3$ and $5 = 6$

Number of integers divisible by $2$ and $3$ and $5 = 3$

Total numbers divisible by $2$ or $3$ or $5 = 50 + 33 + 20 -16 -10 - 6 + 3 = 74$

Total number not divisible by $2$ or $3$ or $5 = 26$

Probability $= 0.26$ [EDIT]

What is the meaning of not divisible by 2,3 or 5?

1. (Not div by 2) or (not div by 3) or ( not div by 5)

2. ( Not div by 2) and ( not div by 3 ) and ( not div by 5)
option 2. ( Not div by 2) and ( not div by 3 ) and ( not div by 5)
In case anyone has doubt whether to take $ceil$ or $floor$, remember we are finding number of numbers divisible by $x$ (say $3$) among a set of $100$ no.s. So $100/3 = 33.33$

If we take $34$ no.s then it’s wrong because we don’t have $34$ numbers. We actually have $33$ only. Hence $floor$ value is always taken.

This is Brute Force method but takes very less time because we just need to check number is not div by 2,3 or 5.

Total no of possible outcomes N(s) = 100

N(e)=Number's not divisible by (2 OR 3 OR 5)  = (Not Div by 2  AND Not Div by 3 Not Div by 5 )      /// Demargon's law

N(e) = {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89,91,97} = 26

Prob = N(e) / N(s) = 26/100 = 0.26

by

### 1 comment

In examination setting,this takes long time and moreover, using this approach one may not be confident
There are total 100 numbers, out of which

50 numbers are divisible by 2,
33 numbers are divisible by 3,
20 numbers are divisible by 5

Following are counted twice above
16 numbers are divisible by both 2 and 3
10 numbers are divisible by both 2 and 5
6 numbers are divisible by both 3 and 5

Following is counted thrice above
3 numbers are divisible by all 2, 3 and 5

So total numbers divisible by 2, 3 and 5 are = = 50 + 33 + 20 - 16 - 10 - 6 + 3 = 103 - 29 = 74 So probability that a number is number is not divisible by 2, 3 and 5 = (100 - 74)/100 = 0.26

https://math.stackexchange.com/a/1034611

I think this one is a better way to find number of divisors. Please check and confirm if its good or not.