We break down the classic Project Euler challenge: sum all natural numbers below 1000 that are multiples of 3 or 5. We’ll show how a little number theory—arithmetic progressions and inclusion-exclusion—lets you craft a fast Python solution. No brute force here: derive the sums with formulas, subtract duplicates (multiples of 15), and discuss generalizing to other divisor sets. A clean example of turning math into elegant code and what it reveals about algorithms.

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