Number of Zeroes in the factorial of a number:-
Number of zeroes in the factorial of a number will be the highest power of 10 in the factorial. & 10 can be written as 2 × 5 in its prime factor form.
& we know that in any factorial power of 2 is always greater than power of 5. Hence to find the number of trailing zeroes we only need to find the highest power of 5 in the factorial & that will be the number of trailing zeroes in the factorial of the number.
(Q). Find the highest power of 10 or number of trailing zeroes in 160!.
Solution:- 10 = 2 × 5
i.e. highest power of 5 in 160! is : 32 + 6 + 1 = 39
∴ Number of trailing zeroes in 160! is 39 Answer
(Q). Find the highest power of 1000 in 1000!
Solution:- 1000 = 10 × 10 × 10 = 2³ × 5³
now first we will find the highest power of 5 in 1000!
i.e. = 200 + 40 + 8 + 1 = 249
∴ highest power of 5 in 1000! is 249 & that of 5³ will be:-
= 81
∴ Highest power of 1000 in 1000! will be 81 Answer