MHB Number of Divisors: 10 Divisors of 21600 Divisible by 10

[juantheron]

How many divisors of $21600$ are divisible by $10$ but not by $15$?

 

[MarkFL]

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?

 

[juantheron]

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]

I think you are on the right track with the prime factorization. I would write it as:

$$21600=2\cdot5\left(2^4\cdot3^3\cdot5 \right)= 10\cdot2^4\cdot3^3\cdot5$$

Now looking at the factor to the right of 10, consider a divisor of 21600 of the form:

$$2^{n_1}\cdot3^{n_2}\cdot5^{n_3}$$

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?