Question

alexio has $100$ cards numbered $1$-$100$, inclusive, and places them in a box. alexio then chooses a card from the box at random. what is the probability that the number on the card he chooses is a multiple of $2$, $3$ or $5$? express your answer as a common fraction.

Answer 1

37/50

Explanation:

You want the fraction of integers in the range [1, 100] that are divisible by 2, 3, or 5.

Divisibility

Attached is a Venn diagram showing how the divisibility of numbers from 1 to 100 stacks up. Circle A includes all 50 numbers divisible by 2; Circle B counts all 33 numbers divisible by 3; and Circle C counts the 20 numbers divisible by 5.

Where the circles overlap, there are counts of the numbers divisible by the relevant combination of factors. For example, there are 3 numbers divisible by 2, 3, and 5. (They are 30, 60, 90.)

Probability

In all, there are 74 numbers in the range 1–100 that are divisible by 2, 3, or 5.

The probability that a card chosen at random will have a number divisible by 2, 3, or 5 is 74/100 = 37/50.

P( 2, 3, or 5 divides N) = 37/50.

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Additional comment

Roughly, the number of numbers in range [A, B] divisible by n is (B -A)/n. This is how we arrived at the counts shown in the attachment. For example, There are 100/(2·5) = 10 numbers divisible by both 2 and 5. Of those, there are 10/3 = 3 divisible also by 3. So, the number 3 is found in the area ABC, and the number 10-3=7 is found in the area AB'C.

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