Difference between revisions of "2024 SSMO Speed Round Problems/Problem 1"
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Find the sum of the distinct prime factors of <math>2024^2 - 1</math>. | Find the sum of the distinct prime factors of <math>2024^2 - 1</math>. | ||
| − | ==Solution== | + | ==Solution 1== |
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| + | Note that <math>2024^2-1</math> can be factored as <math>(2024-1)(2024+1) = (2023)(2025).</math> Then, <math>2023</math> can be factored as <math>7\cdot17^2</math> and <math>2025</math> can be factored as <math>45^2 = 3^4\cdot5^2.</math> Thus, <cmath>2024^1-1 = 2023\cdot2025 = 3^4\cdot5^2\cdot7\cdot17^2,</cmath> meaning the sum of the distinct prime factors is <math>3+5+7+17 = \boxed{32}.</math> | ||
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| + | ~SMO_Team | ||
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| + | ==Solution 2== | ||
<math>2024^2-1 = (2024-1)(2024+1) = 2023 \cdot 2025 = 7 \cdot 17^2 \cdot 3^4 \cdot 5^2</math>. The sum of the distinct prime factors are <math>3+5+7+17 = \boxed{32}</math>. | <math>2024^2-1 = (2024-1)(2024+1) = 2023 \cdot 2025 = 7 \cdot 17^2 \cdot 3^4 \cdot 5^2</math>. The sum of the distinct prime factors are <math>3+5+7+17 = \boxed{32}</math>. | ||
-Vivdax | -Vivdax | ||
Latest revision as of 14:23, 10 September 2025
Problem
Find the sum of the distinct prime factors of 
.
Solution 1
Note that 
can be factored as 
Then, 
can be factored as 
and 
can be factored as 
Thus, ![\[2024^1-1 = 2023\cdot2025 = 3^4\cdot5^2\cdot7\cdot17^2,\]](http://latex.artofproblemsolving.com/f/3/7/f37d80f658589d06797c6210049ff555f64c9105.png)
meaning the sum of the distinct prime factors is 
~SMO_Team
Solution 2
. The sum of the distinct prime factors are 
.
-Vivdax
