## Task A-1.2.

On the playground, there are 2014 athletes who have numbers from 1 to 2014 on their jerseys (each number is on exactly one jersey). At the beginning, all of them are standing. At certain time intervals, the coach shouts out all natural numbers from 1 to 2014 in sequence. Athletes whose jersey numbers are multiples of the shouted number immediately change their position from standing to squatting or vice versa.

How many athletes are squatting after the coach shouts the number 2014?
## Solution.

Each athlete will change position as many times as their number has divisors.

Thus, at the end, those athletes whose numbers have an odd number of divisors will be in the squat position.

All divisors of a number $n$ can be grouped into two-element sets $\left\{d, \frac{n}{d}\right\}$, except when $n=d^{2}$ for some natural number $d$, in which case instead of one two-element set, we have a one-element set (since $\frac{n}{d}=d$). This means that a natural number has an odd number of divisors if and only if it is a perfect square.

Among the numbers $1, \ldots, 2014$, the squares are $1^{2}, 2^{2}, \ldots, 44^{2}$ (since $44^{2}=1936$ and $45^{2}=2025$).

44 athletes will remain in the squat position.

Note: The statement that a natural number has an odd number of divisors if and only if it is a perfect square can also be proven in the following way. If $n=p_{1}^{k_{1}} \cdots p_{s}^{k_{s}}$, where $p_{i}$ are distinct prime numbers and $k_{i} \geqslant 1$, then $n$ has $\left(k_{1}+1\right) \cdots\left(k_{s}+1\right)$ divisors (the factor $p_{i}$ can appear between 0 and $k_{i}$ times in a divisor). Therefore, the number of divisors of $n$ is odd if and only if all numbers $k_{i}$ are even.