# Number of Divisors

#### jacks

##### Well-known member
How many divisors of $21600$ are divisible by $10$ but not by $15$?

#### MarkFL

Staff member
In order to assist our helpers in knowing just where you are stuck, can you show what you have tried or what your thoughts are on how to begin?

#### jacks

##### Well-known member
Prime factor of $21600 = 2^5 \times 3^3 \times 5^2$

Now No. is Divisible by $10$ If It Contain at least one factor of $5$ and $2$

and No. is Non Divisible If It not Contain at least one $3$ and $5$

Now How Can I proceed after that

Thanks

#### MarkFL

$$\displaystyle 21600=2\cdot5\left(2^4\cdot3^3\cdot5 \right)= 10\cdot2^4\cdot3^3\cdot5$$
$$\displaystyle 2^{n_1}\cdot3^{n_2}\cdot5^{n_3}$$
What are the number of choices we have for the parameters $n_i$ such that this factor is not divisible by 3? Then apply the fundamental counting principle. What do you find?