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Divisibility by small primes

ProbabilityMedium

An integer is chosen uniformly at random from $1$ to $100$ . What is the probability that it is divisible by at least one of $2$ , $3$ or $5$ ?

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Hint

Count multiples of each number, then correct for overlaps.

Solution

Single counts: $50$ multiples of $2$ , $33$ of $3$ , $20$ of $5$ , total $103$ .
Pair overlaps: multiples of $6$ number $16$ , of $10$ number $10$ , of $15$ number $6$ , total $32$ .
Multiples of $30$ number $3$ , and they must be added back.
The union has $103 - 32 + 3 = 74$ numbers.
The probability is $\frac{74}{100} = \frac{37}{50}$ .

Answer: $\frac{37}{50}$