Some sum of U n I = US?

Level 2

If we exchange the positions of just two digits from the lowest common multiple of all positive numbers up to n, we will get the square of the sum of all positive numbers up to n. What is the largest prime factor of the sum between that lowest common multiple and that square?

The answer is 101.

1 solution

Ricardo Moritz Cavalcanti
Dec 24, 2020

Using a calculator, I found that the lowest number satisfying the condition of the problem is $n = 9$ :

lcm $(1,2,...,9) = 2025$

$(1+2+...+9)^2 = 45^2 = 2520$

$2025 + 2520 = 4545 = 3^2\cdot 5\cdot 101$

The answer, therefore, is $\mathbf{101}$ .

Hi, I just saw your name at the comment section of Quanta Magazine, Conway puzzle. Thanks for giving this a try (nowhere near as fun as a Conway's, though) and posting the solution!

Saya Suka - 5 months, 2 weeks ago

Not a brilliant solution, though, just trial and error. :-)

Ricardo Moritz Cavalcanti - 5 months, 2 weeks ago

Some sum of U n I = US?

The answer is 101.

1 solution

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