$\underline{\text { Akulichi I.F. }}$

Do there exist ten pairwise distinct natural numbers such that their arithmetic mean is

a) exactly six times;

b) exactly five times;

their greatest common divisor?
a) For example, $1, 2, \ldots, 9, 15$. The sum of these numbers is 60, the arithmetic mean is 6, and the GCD is 1.

b) Let the GCD of ten numbers $a_{1}<a_{2}<\ldots<a_{10}$ be $d$. Then $a_{1} \geq d, a_{2} \geq 2 d, \ldots, a_{10} \geq 10 d$. Therefore, the sum of these numbers is not less than $55 d$, and the arithmetic mean is not less than $5.5 d$.

## Answer

a) They exist; b) they do not exist.