Task A-1.4. (8 points)

How many divisors does the number 288 have?
## Solution.

Let's factorize the number 288:

$$
288=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 2^{5} \cdot 3^{2}
$$

First method.

A divisor of 288 cannot have prime factors other than 2 and 3.

The factor 2 can appear 0, 1, 2, 3, 4, or at most 5 times, giving us 6 possibilities.

For the appearance of the factor 3, there are three possibilities: $3^{0}, 3^{1}, 3^{2}$.

The total number of divisors is $6 \cdot 3=18$.

Second method.

The divisors are: $1,2,3,4,6,8,9,12,16,18,24,32,36,48,72,96,144$ and 288. There are 18 of them.

For 4 points: minor error in counting.

For 8 points: correct procedure and result, or all divisors listed correctly (without the result "18").