## Task 4 - 220624

For five different natural numbers $a, b, c, d, e$ that are distinct from each other and from 0, the following eight requirements are imposed:

(1) $a$ is an integer multiple of $e$,

(2) $b$ is a divisor of $c$,

(3) $c$ is an integer multiple of $e$,

(4) $d$ is a divisor of $e$,

(5) $a$ is an integer multiple of $b$,

(6) $b$ is a divisor of $d$,

(7) $c$ is an integer multiple of $a$,

(8) $a$ is an integer multiple of $d$.

Investigate whether these eight requirements can be satisfied and whether they determine the order of the five numbers in terms of their size!

If this is the case, state this order; start with the largest of the five numbers!
From (7) it follows that $c>a$,

from (1) it follows that $a>e$,

from (4) it follows that $e>d$,

from (6) it follows that $d>b$.

Therefore, only with the arrangement $c>a>e>d>b$ can the requirements (1) to (8) be satisfied.

They are satisfiable, for example, by $b=1, d=2, e=4, a=8, c=16$; because 8 is an (integer) multiple of 4, 1 is a divisor of 16, 16 is an (integer) multiple of 4, 2 is a divisor of 4, 8 is an (integer) multiple of 1, 1 is a divisor of 2, 16 is an (integer) multiple of 8, and 8 is an (integer) multiple of 2.

Solutions of the II. Round 1982 taken from [5]

### 3.24 XXIII. Olympiad 1983

### 3.24.1 I. Round 1983, Class 6