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Lesson 7 of 8

Enumerate overlapping conditions

Use subset parity and LCM to count a union of divisibility conditions.

Understand the idea

General inclusion-exclusion adds intersections of odd numbers of sets and subtracts intersections of even numbers. A nonempty mask selects one intersection. For numbers divisible by every selected divisor, use their LCM; the intersection count in [1, N] is floor(N/LCM).

Trace a small example

Up to 20, multiples of 2 number 10 and multiples of 3 number 6. Their overlap is multiples of 6, numbering 3. The union therefore has 10 + 6 − 3 = 13. Divisors 2 and 4 have intersection multiples of 4, not multiples of 8.

Conditions and cost

LCM, not an unproved product, handles shared prime factors. Cap the LCM at N before multiplying; if it exceeds N, that intersection contributes zero. With k conditions, the straightforward method visits 2^k − 1 masks and needs small k.

Pause & predict

Numbers divisible by both 4 and 6 are exactly the multiples of which value?

Choose an answer
Try it yourself

Turn the idea into code

Count numbers from 1 through 30 divisible by 3 or 5.

Give me a hint

Subtract the overlap at multiples of 15.

Show the explained solution

floor(30/3) + floor(30/5) − floor(30/15) = 10 + 6 − 2 = 14.

Your notes

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