Dr. Troll 2

Number Theory Level 2

When all the divisors of x x are summed, the result is x + 1 x+1 . Find the sum of the ten smallest positive values of x x .


The answer is 129.

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Alex Segesta by Alex Segesta , 20, USA

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5 solutions

Alex Segesta
May 10, 2014

A prime number has only two divisors: 1 and itself. We just sum the first 10 positive prime numbers and get 129 \boxed{129} .

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Can you prove that primes are the only numbers that satisfy this?

Daniel Liu - 7 years, 1 month ago

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If x is not a prime number then its divisors will include: x,1 and other number(s) whose sum is, say y.

When the divisors are all added, we get x+y+1

Meenakshi Janardanan - 7 years ago

Hm, if you pick any positive integer x x , then it will have at least 2 factors: 1 and itself. Therefore, the minimum possible sum is x + 1 x+1 aka prime with an exception of 1.

Alex Segesta - 7 years, 1 month ago

Just a question, I hope someone can clarify. When you say divisors, does it ALWAYS mean the positive divisors only? Because strictly speaking, -1 is also a divisor of every number. Therefore, shouldn't you say the "positive divisors of x"?? Don't get me wrong, the problem was really good, and I did indeed use your method (of determining all the primes and summing them), but that little bit I just wanted to clarify. Thanks.

Kevin Mano - 6 years, 10 months ago

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If negative divisors are included, it would make every sum zero, which makes the question having only one solution: -1.

Kenny Lau - 5 years, 10 months ago
Gautam Sharma
Jul 22, 2014

primes from 2 to 29 (inclusive).

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wow....shortest ever explaination

Anchal Rajawat - 6 years, 9 months ago
Edwin Gray
Feb 16, 2019

If the sum of x and 1 is x + 1, then x is prime. The sum of the first 10 primes is 129

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Vardaan Chopra
Jan 28, 2015

x must be a prime number. Starting from smallest - 2+3+5+7+11+13+17+19+23+29 =129

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Trajan Hammonds
Jul 25, 2014

In order for a number x to have divisors which sum to x+1, the number has to be prime, meaning the two divisors are x and 1. So we simply add the first 10 prime numbers.

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