GATE CS 2017 Set 1 Q.47 (CS Q.47): Combinatorics, Counting, Recurrence Relations and Generating Functions

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ______.

Your answer

Official answer and solution

Solution

Apply the Principle of Inclusion-Exclusion to count integers between 1 and 500 divisible by 3, 5, or 7.

Let $U = \{1, 2, \dots, 500\}$ , so $|U| = 500$ .
Let $A, B, C$ be the subsets of $U$ containing integers divisible by 3, 5, and 7 respectively.

Calculate the size of each subset:

|A| = \lfloor 500 / 3 \rfloor = 166

|B| = \lfloor 500 / 5 \rfloor = 100

|C| = \lfloor 500 / 7 \rfloor = 71

Calculate the pairwise intersections using least common multiples:

Calculate the three-way intersection:

\text{lcm}(3, 5, 7) = 105 \implies |A \cap B \cap C| = \lfloor 500 / 105 \rfloor = 4

Applying the Principle of Inclusion-Exclusion:

|A \cup B \cup C| = (|A| + |B| + |C|) - (|A \cap B| + |A \cap C| + |B \cap C|) + |A \cap B \cap C|

|A \cup B \cup C| = (166 + 100 + 71) - (33 + 23 + 14) + 4 = 337 - 70 + 4 = 271

Final answer

271.0

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