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:

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

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?

 

[blue_raver22]

I would approach this problem by first understanding the concept of divisors and their relationship to prime factorization. Divisors are numbers that can evenly divide a given number without leaving a remainder. In this case, the number of divisors of 21600 is 10, meaning there are 10 numbers that can evenly divide 21600 without leaving a remainder.

Next, I would use prime factorization to break down 21600 into its prime factors, which are 2, 3, and 5. This can be represented as 2^5 * 3^3 * 5^2. From this, we can see that the divisors of 21600 are combinations of these prime factors.

To answer the question of how many divisors of 21600 are divisible by 10 but not by 15, we need to consider the prime factors of 10 and 15. The prime factors of 10 are 2 and 5, while the prime factors of 15 are 3 and 5.

Since a number is divisible by another number if it contains all of its prime factors, we can see that the divisors of 21600 that are divisible by 10 must contain at least one 2 and one 5. Additionally, the divisors that are not divisible by 15 must not contain any 3.

Using this information, we can list out the divisors of 21600 that are divisible by 10 but not by 15 as follows:

1) 2^1 * 5^1 = 10
2) 2^2 * 5^1 = 20
3) 2^3 * 5^1 = 40
4) 2^4 * 5^1 = 80
5) 2^5 * 5^1 = 160

Therefore, there are 5 divisors of 21600 that are divisible by 10 but not by 15. This can also be confirmed by counting the number of divisors with prime factors of 2 and 5, which is also 5.

In conclusion, the number of divisors of 21600 that are divisible by 10 but not by 15 is 5.