2003 AMC 10A Problem 15

P11-15~2 minPrint

Divisibility & Factors Basic Probability Count the complement and subtract from the total

What is the probability that an integer in the set $\{1,2,3,...,100\}$ is divisible by $2$ and not divisible by $3$ ?

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Key insight

Count the 50 even numbers, then take away the even numbers that are also multiples of 3, which are the 16 multiples of 6. That leaves 34.

Solution

Start with all the even numbers, then remove the ones that are also multiples of $3$ .

1. Even numbers from $1$ to $100$ : $2, 4, 6, \ldots, 100$ . There are $100 \div 2 = 50$ of them.
2. An even number that is also a multiple of $3$ is a multiple of $6$ . Those are $6, 12, 18, \ldots$ . Since $6 \times 16 = 96$ and $6 \times 17 = 102$ is too big, there are $16$ of them.
3. Even numbers that are not multiples of $3$ : $50 - 16 = 34$ .
4. The probability is
$\frac{34}{100} = \frac{17}{50}.$

The answer is $\boxed{\textbf{(C)}\ \frac{17}{50}}$ .

Why this works

"Divisible by $2$ but not by $3$ " means: take the multiples of $2$ , then throw out the ones that are multiples of both, which are the multiples of $6$ . The count of multiples of $6$ up to $100$ must be a whole number, $16$ , so the answer is not exactly one third of the evens. Whenever the upper limit is not a multiple of the step, count carefully instead of using a fraction.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Multiples of $2$ up to $100$ : $\lfloor 100/2 \rfloor = 50$ .

Among these, the ones also divisible by $3$ are the multiples of $\operatorname{lcm}(2,3) = 6$ : $\lfloor 100/6 \rfloor = 16$ (namely $6, 12, \dots, 96$ ).

Even numbers that are not multiples of $3$ : $50 - 16 = 34$ . The probability is $\dfrac{34}{100} = \dfrac{17}{50}$ .

The answer is $\boxed{\textbf{(C)}\ \frac{17}{50}}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Introduction to Counting & Probability, Chapter 7: Introduction to Probability
Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques (§2.3)
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)