An integer n is selected at random from the set \{1, 2, 3, \dots, 500\}. What is the proba… — GMAT

An integer $n$ is selected at random from the set $\{1, 2, 3, \dots, 500\}$ . What is the probability that $n$ is a perfect square that leaves a remainder of $1$ when divided by $3$ ?

Answer: 0.03

Answer

0.03

The sample space consists of 500 equally likely outcomes. Perfect squares in this range are generated by integers

k

from 1 to 22. Evaluating

k^2 \pmod 3

shows that

k^2

has a remainder of 1 when divided by 3 for all integers

k

not divisible by 3. Excluding the 7 multiples of 3 from the 22 possible values of

k

leaves 15 favorable integers. Dividing the 15 favorable outcomes by the 500 total outcomes gives a probability of 0.03.

Step-by-Step Solution

Determine total possible outcomes.

The total number of integers in the set

\{1, 2, 3, \dots, 500\}

N = 500

Each integer in the set is equally likely to be selected.

Find the range of square roots for perfect squares in the set.

Since

22^2 = 484 \le 500

and

23^2 = 529 > 500

n = k^2

for

k \in \{1, 2, \dots, 22\}

There are 22 perfect squares in the given sample space.

Apply remainder constraints to

k^2

A square

k^2

leaves a remainder of

1

when divided by

3

if and only if

k

is not divisible by

3

k

is a multiple of 3,

k^2

is divisible by 3 (remainder 0). If

k

is not a multiple of 3,

k \equiv 1

2 \pmod 3

, so

k^2 \equiv 1 \pmod 3

Count the number of favorable outcomes.

The multiples of 3 up to 22 are

3, 6, 9, 12, 15, 18, 21

(7 numbers). Thus, there are

22 - 7 = 15

values of

k

that satisfy the condition.

Favorable outcomes correspond exactly to integers

k \in \{1, 2, \dots, 22\}

that are not multiples of 3.

Compute the single-event probability.

Probability

= \frac{15}{500} = 0.03

The probability of a single event is the number of favorable outcomes divided by the total number of outcomes.

Key Concept

Basic Single-Event Probability with Number Properties

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