How many consecutive zeros are there at the end of the integer obtained in the product $1^2 \times 2^4…
How many consecutive zeros are there at the end of the integer obtained in the product $1^2 \times 2^4 \times 3^6 \times 4^8 \times \cdots \times 25^{50}$?
50
55
100
200
Solution
A trailing zero comes from a factor of 10 = 2 x 5. The number of 5s is fewer, so count the 5s. The terms contributing 5s are $5^{10}$ (10 fives), $10^{20}$ (20 fives), $15^{30}$ (30 fives), $20^{40}$ (40 fives) and $25^{50}$ which has $25 = 5^2$ giving $5^{100}$ (100 fives). Total $= 10 + 20 + 30 + 40 + 100 = 200$ zeros.