21 votes

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

30 votes

Best answer

Answer - $0.26$

no of integers divisible by $2 = 50$

no of integers divisible by $3 = 33$

no of integers divisible by $5 = 20$

no of integers divisible by $2$ and $3$ = $16$

no of integers divisible by $2$ and $5$ = $10$

no of integers divisible by $3$ and $5$ = $6$

no 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]

no of integers divisible by $2 = 50$

no of integers divisible by $3 = 33$

no of integers divisible by $5 = 20$

no of integers divisible by $2$ and $3$ = $16$

no of integers divisible by $2$ and $5$ = $10$

no of integers divisible by $3$ and $5$ = $6$

no 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]

5 votes

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

5 votes

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

0 votes

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.