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As some nations like to say "Heavy theorems mostly do not help"

Source: European Mathematical Cup 2022, Senior Division, Problem 2

December 20, 2022

number theoryDivisorsgreatest common divisorDivisibility

Problem Statement

We say that a positive integer

n

is lovely if there exist a positive integer

k

and (not necessarily distinct) positive integers

d_1

d_2

\ldots

d_k

such that

n = d_1d_2\cdots d_k

and

d_i^2 \mid n + d_i

for

i=1,2,\ldots,k

.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than

1

, which is a perfect square of an integer?