An integer \(N\) is chosen at random with \(1 \leq N \leq 100 .\) What is the probability that \(N\) is divisible by \(11 ?\) That \(N>90 ?\) That \(N \leq 3 ?\) That \(N\) is a perfect square?

In mathematics, divisibility refers to the ability of one integer to be divided by another without leaving a remainder. For an integer to be divisible by another, the division of these two numbers should result in a whole number.
For example, 15 is divisible by 5 because when you divide 15 by 5, you get 3, which is a whole number.
Divisibility plays a significant role in various fields, including number theory, where it helps in determining prime numbers and in solving equations.
In our exercise, the numbers from 1 to 100 divisible by 11 are: 11, 22, 33, 44, 55, 66, 77, 88, and 99.
To find if a number is divisible by 11 among any set of integers, observe if the numbers perfectly divide without a remainder.

Random integers are numbers chosen in such a way that every possible number in a given set has an equal chance of being selected. This process is crucial in probability and statistics for creating unbiased samples.
For the given exercise, we look at integers randomly chosen from 1 to 100. There is no bias in selection, meaning the integer could just as likely be 50 as it could be 1.
This randomness helps in calculating the probability of certain conditions, like being divisible by 11 or greater than 90.
The key aspect of randomness is that it assures every number in the defined range has equal opportunity to be selected, ensuring a fair analysis in probability calculations.

Perfect squares are numbers that result from squaring a whole number. For example, 25 is a perfect square because it is the result of 5 times 5.
Perfect squares are important in various areas of mathematics, including algebra and geometry. They simplify complex equations and can represent areas of squares in geometric problems.
In our exercise, the perfect squares between 1 and 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Spotting perfect squares within a range helps in solving probability problems where the condition involves numbers that are perfect squares.
It’s essential to know that the intervals between perfect squares grow larger as the numbers get bigger, which can sometimes make identifying them a bit more challenging.

Mathematical probability is the measure of the likelihood that an event will occur. In probability theory, we often use calculations to determine this likelihood.
To find the probability of a specific event, we divide the number of successful outcomes by the total number of possible outcomes.
In the given exercise, we calculate the probability of different scenarios such as a number being divisible by 11 or greater than 90.
The formula used for these calculations is simple: \ \[ P(Event) = \frac{Number \ of \ Successful \ Outcomes}{Total \ Number \ of \ Possible \ Outcomes} \ \]
For example, the probability that a number is greater than 90 is given by: \ \[ \frac{10}{100} = 0.10 \] Where 10 is the count of numbers greater than 90.
Understanding this basic probability formula helps solve a variety of problems involving chance and likelihood.