I choose a number uniformly at random from the range...

I choose a number uniformly at random from the range $[1,1,000,000]$. Using the inclusion-exclusion principle, determine the probability that the number chosen is divisible by one or more of 4,6, and $9 .$

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Key Concepts

Divisibility and Least Common Multiple (LCM)

Divisibility criteria are used to determine if a number can be evenly divided by another number without leaving a remainder. When counting numbers divisible by several integers, it is important to use the least common multiple (LCM) to identify numbers that are multiples of more than one divisor, which is essential for calculating intersections in the inclusion-exclusion principle.

Uniform Probability Distribution

A uniform probability distribution assigns equal likelihood to each outcome in a finite set. When a number is chosen uniformly at random from a given range, every number within that range has the same probability of being selected, allowing probabilities to be computed by dividing the count of favorable outcomes by the total number of possible outcomes.

Inclusion-Exclusion Principle

The inclusion?exclusion principle is a combinatorial method used to compute the cardinality of the union of multiple sets by summing the sizes of individual sets, then subtracting the sizes of their pairwise intersections, adding the sizes of triple intersections, and so on. This technique helps in accurately counting elements that satisfy at least one of several properties without double counting those that satisfy multiple properties.

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