All the positive integers less than 49 are multiplied together. How many zeros will this product end in?
We just need to count how many factors of 5 there are in all those numbers.
Why?
1. Zeros on the end of this product are produced by and only by factors of 10.
2. Every pair of factors 5x2 will be a factor of 10.
3. There are obviously more factors of 2 than 5, since every even number
contributes one or more factors of 2.
4. Therefore, the number of 10 factors is the same as the number of 5 factors,
for each factor of 5 paired with a factor of 2 will contribute a factor of
10.
Every multiple of 5 produces 1 factor of 5 except 25, which produces 2 factors
of 5. Therefore among these 9 numbers 5,10,15,20,25,30,35,40,45 there will
be 10 factors of 5, with the 25 contributing an extra factor of 5
Answer: There will be 10 zeros on the end of the product.
Edwin
