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2024 SSMO Speed Round Problems/Problem 1

Problem

Find the sum of the distinct prime factors of $2024^2 - 1$.

Solution 1

Note that $2024^2-1$ can be factored as $(2024-1)(2024+1) = (2023)(2025).$ Then, $2023$ can be factored as $7\cdot17^2$ and $2025$ can be factored as $45^2 = 3^4\cdot5^2.$ Thus, \[2024^1-1 = 2023\cdot2025 = 3^4\cdot5^2\cdot7\cdot17^2,\] meaning the sum of the distinct prime factors is $3+5+7+17 = \boxed{32}.$

~SMO_Team

Solution 2

$2024^2-1 = (2024-1)(2024+1) = 2023 \cdot 2025 = 7 \cdot 17^2 \cdot 3^4 \cdot 5^2$. The sum of the distinct prime factors are $3+5+7+17 = \boxed{32}$.

-Vivdax

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