An integer $n$ is called apocalyptic if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$. Find the smallest positive apocalyptic number.
Problem
Source: Rioplatense Olympiad 2012, Level 3, Problem 1
Tags: number theory, prime factorization, number theory proposed
dhwang314
27.08.2014 02:13
Prime factorization of 3528.
$3528=2^3*3^2*7^2$. We are using the largest power of 2 as possible to get the power of 2 is 5. Using the sum of divisors trick, you get that 2^5+2^4...2^0 is 63. This leaves us with 8*7 left to fill. Now, fill in 3 and 13 to get 39*32=78+1170. The answer is 1248.
mathocean97
27.08.2014 07:04
@above, please read the question more carefully. The problem does not state that the sum of the divisors is 3528, instead, it states that 6 of its divisors sum to 3528. Instead, we just do the following: let the 6 divisors be $d_1, d_2, d_3, d_4, d_5, d_6$ be divisors of $d$. Then $\sum_{i = 1}^d d_i \le d+d/2+d/3+d/4+d/5+d/6 = 49d/20$. Therefore, $49d/20 \ge 3528 \implies d \ge 1440$. $1440$ works ofc by the previous: $1440/1+1440/2+1440/3+1440/4+1440/5+1440/6 = 3528$.
dhwang314
18.09.2014 00:51
Yeah, I noticed. Good job!
BR1F1SZ
15.02.2025 15:52
This is (almost) the same problem as ISL 2022 N1!