We are tasked with finding how many divisors of 14,700 are divisible by 10 but not by 100. Let's break this down step by step:

Step 1: Prime factorization of 14,700

We begin by finding the prime factorization of 14,700.

$14,700 = 2^2 \times 3 \times 5^2 \times 7^2$

Step 2: Divisors divisible by 10

A number divisible by 10 must include at least $2^1 \times 5^1$ in its prime factorization (since $10 = 2 \times 5$).

To find divisors of 14,700 divisible by 10, we must focus on the divisors that contain at least one factor of 2 and one factor of 5.

Step 3: Divisors divisible by 100

A number divisible by 100 must include at least $2^2 \times 5^2$ in its prime factorization (since $100 = 2^2 \times 5^2$).

We need to find the divisors that are divisible by 10 but not by 100, meaning we want divisors that contain at least one factor of 2 and one factor of 5, but not both $2^2$ and $5^2$.

Step 4: Counting the divisors divisible by 10 but not 100

We now break down the counting:

1. Divisors divisible by 10:
   A divisor divisible by 10 must include $2^1 \times 5^1$.
   Consider the prime factorization $2^2 \times 3 \times 5^2 \times 7^2$.
   A divisor divisible by 10 can take:

   • $2^1$ or $2^2$ (two choices for the power of 2),
   • $5^1$ or $5^2$ (two choices for the power of 5),
   • Any power of 3: $3^0$ or $3^1$ (two choices),
   • Any power of 7: $7^0$, $7^1$, or $7^2$ (three choices).

   So the total number of divisors divisible by 10 is:

   $2 \times 2 \times 2 \times 3 = 24$

2. Divisors divisible by 100:
   A divisor divisible by 100 must include $2^2 \times 5^2$.
   For divisors divisible by 100, we have:

   • Exactly $2^2$ (one choice),
   • Exactly $5^2$ (one choice),
   • Any power of 3: $3^0$ or $3^1$ (two choices),
   • Any power of 7: $7^0$, $7^1$, or $7^2$ (three choices).

   So the total number of divisors divisible by 100 is:

   $1 \times 1 \times 2 \times 3 = 6$

Step 5: Subtract to find divisors divisible by 10 but not 100

The number of divisors divisible by 10 but not by 100 is:

$24 - 6 = 18$

Thus, the number of divisors of 14,700 that are divisible by 10 but not by 100 is 18.

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. What is the number of divisors of 14,700 in total?
2. How would the process change if we were looking for divisors divisible by 5 instead of 10?
3. Can you find how many divisors of 14,700 are divisible by 7?
4. What if we wanted divisors divisible by 20 but not by 200?
5. How would you approach finding the sum of all divisors of 14,700?

Tip: When solving divisor problems, prime factorization is essential for breaking down the conditions and constraints efficiently.