If you multiply 100!*200!*300!, how many 0s are there?
To find the number of zeros at the end of a product like 100!*200!*300!, we need to determine how many times the number 10 can be factored out.
Firstly, let's consider the prime factors of 10. We know that 10 = 2 * 5. A zero is formed when a multiple of 10 is multiplied. Since 10 can be factored as 2 * 5, we need to count the number of 2's and 5's in the product.
In a factorial, there are always more multiples of 2 than multiples of 5. So, we will count the number of multiples of 5 in the product, as that will determine the number of zeros.
To find the number of multiples of 5, we can divide each number (100, 200, and 300) by 5 and sum up the quotients. However, we need to consider the multiples of 25 (5^2) and 125 (5^3) as they introduce additional factors of 5.
Let's calculate it step by step:
1) Counting the multiples of 5:
- In 100!, there are 100/5 = 20 multiples of 5.
- In 200!, there are 200/5 = 40 multiples of 5.
- In 300!, there are 300/5 = 60 multiples of 5.
2) Counting the multiples of 25 (5^2):
- In 100!, there are 100/25 = 4 multiples of 25.
- In 200!, there are 200/25 = 8 multiples of 25.
- In 300!, there are 300/25 = 12 multiples of 25.
3) Counting the multiples of 125 (5^3):
- In 100!, there are 100/125 = 0 multiples of 125.
- In 200!, there are 200/125 = 1 multiples of 125.
- In 300!, there are 300/125 = 2 multiples of 125.
Adding up all these counts, we get:
20 + 40 + 60 + 4 + 8 + 12 + 0 + 1 + 2 = 147
Therefore, there are 147 zeros at the end of the product 100!*200!*300!.