Friendly Numbers

Futility Closet reports on an intriguing mathematical property called friendly numbers. When you sum the divisors of 30 and divide by 30 itself, you get twelve-fifths. Do the same with 140 and you get the same result, making them friends. Other known pairs include 2480, 6200, and 40640. But here's the puzzle: no one knows whether 10 has a friendly partner—if it does, its smallest friend would be at least ten to the thirtieth power.

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