Project Euler #1: Multiples of 3 and 5

Fri, 10 Apr 2026 00:00:00 +0000

Problem

Find the sum of all multiples of 3 or 5 below 1000.

A simple loop works, of course:

print(sum(x for x in range(1000) if x % 3 == 0 or x % 5 == 0))

But the point of Project Euler isn’t the answer — it’s finding the clean mathematical structure underneath.

Let $S(k, n)$ denote the sum of all multiples of $k$ strictly below $n$. Then:

$$S(k, n) = k \cdot \sum_{i=1}^{\lfloor (n-1)/k \rfloor} i = k \cdot \frac{m(m+1)}{2}$$

where $m = \lfloor (n-1)/k \rfloor$.

Sat, 11 Apr 2026 00:00:00 +0000

Each new term in the Fibonacci sequence is generated by adding the previous two terms. Starting with 1 and 2, the first 10 terms are:

$$1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots$$

Find the sum of even-valued terms whose values do not exceed 4 million.

Call the Fibonacci sequence $F_1 = 1, F_2 = 2, F_3 = 3, \ldots$.

Looking at the parity pattern: odd, even, odd, odd, even, odd, odd, even, $\ldots$