For the number 1440, find the number of factors which are divisible by 5.

To find the number of factors of 1440 that are divisible by 5, follow these steps:

1. Prime Factorization of 1440: First, we need to find the prime factorization of 1440. $1440 = 2^5 \times 3^2 \times 5$

2. Formulating Factors: Any factor of 1440 can be written in the form $2^a \times 3^b \times 5^c$, where $0 \leq a \leq 5$, $0 \leq b \leq 2$, and $0 \leq c \leq 1$.

3. Factors Divisible by 5: For a factor to be divisible by 5, $c$ must be at least 1 (since $5^0 = 1$ is not divisible by 5). Therefore, $c = 1$.

4. Counting the Factors: With $c = 1$, we need to count the combinations of $a$ and $b$:

   • $a$ can take any value from 0 to 5 (6 choices).
   • $b$ can take any value from 0 to 2 (3 choices).

   Thus, the total number of factors is: $6 \times 3 = 18$

Therefore, the number of factors of 1440 that are divisible by 5 is $18$.