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Number of Divisors

jacks

Well-known member
Apr 5, 2012
226
How many divisors of $21600$ are divisible by $10$ but not by $15$?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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
Apr 5, 2012
226
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

Administrator
Staff member
Feb 24, 2012
13,775
I think you are on the right track with the prime factorization. I would write it as:

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

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

\(\displaystyle 2^{n_1}\cdot3^{n_2}\cdot5^{n_3}\)

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?