For the number 1440, find the number of factors which are divisible by 5.12141618None of the above

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

1. Prime Factorization of 1440:

   • 1440 can be factorized as: $1440 = 2^4 \times 3^2 \times 5$

2. Factors of 1440:

   • A factor of 1440 can be written in the form $2^a \times 3^b \times 5^c$, where $0 \leq a \leq 4$, $0 \leq b \leq 2$, and $0 \leq c \leq 1$.

3. Factors Divisible by 5:

   • For a factor to be divisible by 5, $c$ must be 1 (since $5^0 = 1$ is not divisible by 5).
   • Therefore, the factors are of the form $2^a \times 3^b \times 5$.

4. Counting the Factors:

   • $a$ can take any value from 0 to 4 (5 choices).
   • $b$ can take any value from 0 to 2 (3 choices).
   • Since $c$ is fixed at 1, there is only 1 choice for $c$.

5. Total Number of Factors:

   • The total number of factors divisible by 5 is the product of the number of choices for $a$, $b$, and $c$: $5 \times 3 \times 1 = 15$

Therefore, the number of factors of 1440 that are divisible by 5 is $15$.