How many zeros are at the end of the product of the first fifty
positive integers?
To find the number of zeros at the end of the product of the first fifty positive integers, we need to determine the number of factors of 10 in the product. This is because a zero at the end of the number is formed by multiplying 10, which is 2 * 5.
We can count the number of factors of 2 and 5 in the product to determine the number of factors of 10.
Let's break down the product of the first fifty positive integers:
1 * 2 * 3 * 4 * ... * 50
To count the number of factors of 2, we need to identify the number of multiples of 2 in the product. We can do this by dividing each number in the product by 2 until we get a number less than 2:
2/2 = 1
4/2 = 2
6/2 = 3
...
48/2 = 24
50/2 = 25
There are 25 multiples of 2 in the product.
To count the number of factors of 5, we need to identify the number of multiples of 5 in the product. We can do this by dividing each number in the product by 5 until we get a number less than 5:
5/5 = 1
10/5 = 2
15/5 = 3
...
45/5 = 9
50/5 = 10
There are 10 multiples of 5 in the product.
Since we need both a factor of 2 and a factor of 5 to form a factor of 10, we will take the smaller of the two counts. In this case, the count of multiples of 5 (10) is smaller than the count of multiples of 2 (25).
Therefore, we can conclude that there are 10 zeros at the end of the product of the first fifty positive integers.