## Task A-2.1.

Which number has more divisors in the set of natural numbers, $2013^{2}$ or 20480?
## Solution.

The prime factorization of the number 2013 is $3 \cdot 11 \cdot 61$.

1 point

Thus, the divisors of the number $2013^{2}=3^{2} \cdot 11^{2} \cdot 61^{2}$ are of the form $3^{a} \cdot 11^{b} \cdot 61^{c}$ where $a, b, c \in\{0,1,2\}$.

There are $3 \cdot 3 \cdot 3=27$ such numbers.

2 points

The prime factorization of the number 20480 is $20480=2^{12} \cdot 5$. Its divisors are numbers of the form $2^{k}$ and $2^{k} \cdot 5$ for $k=0,1, \ldots, 12$. There are a total of $2 \cdot 13=26$ such numbers.

1 point

The number $2013^{2}$ has more divisors.

1 point