How many zeros are at the end of the number given by the product of all the integers from 10 to 19 inclusive? Please show working.

To determine how many zeros are at the end of the product of all integers from 10 to 19 inclusive, we need to find the number of times 10 is a factor in this product. Each 10 is made up of factors 2 and 5. Therefore, we need to count the number of pairs of factors 2 and 5 in the product.

First, let's write the product explicitly:

\[ 10 \times 11 \times 12 \times 13 \times 14 \times 15 \times 16 \times 17 \times 18 \times 19 \]

To find the number of trailing zeros, we need to count the number of factors of 5 and the number of factors of 2. Since there are generally fewer factors of 5 than factors of 2 in a product, the number of trailing zeros will be determined by the number of factors of 5.

### Counting Factors of 5
We count the number of multiples of 5 in the range from 10 to 19.

The multiples of 5 in this range are:
\[ 10, 15 \]

Each of these contributes at least one factor of 5. Therefore, there are 2 factors of 5.

### Counting Factors of 2
Now, let's count the factors of 2 in the same range:

- 10 has one factor of 2.
- 12 has two factors of 2.
- 14 has one factor of 2.
- 16 has four factors of 2.
- 18 has one factor of 2.

Adding these up, we get:
\[ 1 (from \, 10) + 2 (from \, 12) + 1 (from \, 14) + 4 (from \, 16) + 1 (from \, 18) = 9 \]

There are 9 factors of 2.

### Conclusion
Since the number of trailing zeros is determined by the number of pairs of factors of 2 and 5, and we have 2 factors of 5 and 9 factors of 2, the number of trailing zeros is:

\[ \min(2, 9) = 2 \]

Thus, the product of all integers from 10 to 19 inclusive ends with 2 zeros.

Would you like further details or have any questions about this problem?

Here are 5 more questions you can ask:
1. How do you find the number of trailing zeros in factorials?
2. What is the significance of prime factorization in determining trailing zeros?
3. Can you explain how to count factors of a prime in a range of numbers?
4. How do you determine the number of trailing zeros in a power of a number?
5. What are other methods to determine the number of trailing zeros in a product?

Tip: When solving for trailing zeros, always look for pairs of prime factors 2 and 5, since each pair forms a 10.

Learn how to determine the number of trailing zeros at the end of a product of integers from 10 to 19. This problem involves understanding factorial properties, prime factorization, and counting factors. Suitable for Grades 7-9 students.

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