Find the number of positive integers ≤1976 and divisible by:...

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Key Concepts

Inclusion-Exclusion Principle

This principle is used in combinatorics to correctly count the number of elements that satisfy at least one of several conditions. Instead of simply adding the counts of individual properties—which would double-count the elements satisfying multiple properties—the principle subtracts the counts of pairwise intersections and adds back the counts of three-way intersections, ensuring that every element is counted exactly once.

Divisibility and Multiples Counting

Counting the number of multiples of a given integer within a range is a fundamental technique in number theory and discrete mathematics. This is typically done by dividing the upper bound of the range by the divisor and taking the floor of the result. This method provides a systematic approach to determine how many numbers within the range are divisible by the given divisor.

Least Common Multiple (LCM)

The least common multiple of two or more integers is the smallest positive integer that is a multiple of each number. In problems involving divisibility by multiple numbers, the LCM is used to identify the intersections between divisibility sets. For example, to find numbers divisible by both two divisors, one would count multiples of their LCM, which corresponds to the intersection of the sets of multiples.

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