Subject (2). Consider the following natural numbers:

$$
a=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29 \cdot 31 \text { and } b=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29
$$

a) Prove that the number $a$ is divisible by 2015.

b) Find the largest natural number $n$ such that the number $a+b$ is divisible by $10^n$.

c) Determine how many divisors that are perfect squares the number $a$ has.

Traian Preda
Solution. We have $2015=5 \cdot 13 \cdot 31$.

a) All three prime factors of 2015 are among the factors of $a$.
b) $a+b=32 b=2^{5} \cdot 5^{4} \cdot c$, where $c$ is coprime with 10; it follows that $\max n=4$.

c) We also have $c=3^{8} \cdot 7^{2} \cdot d$, where $d$ is coprime with 21 and square-free, so $a=3^{8} \cdot 5^{4} \cdot 7^{2} \cdot e$, where $e$ is coprime with 105 and square-free. Therefore, there are 30 perfect square divisors.

Stating in the problem that the numbers $a$ and $b$, given by their exact values, are natural, is completely redundant; moreover, the form in which they are given is cumbersome - it would have been more natural to give $b$, and then $a=31 b$, but perhaps it was considered that this would help the competitors factorize $a+b$!?