算法设计与应用基础系列16



Factorial Trailing Zeroes

Given an integer n, return the number of trailing zeroes in n!.

Note: Your solution should be in logarithmic time complexity.

Credits:
Special thanks to @ts for adding this problem and creating all test cases.

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The idea is:

1. The ZERO comes from 10.
2. The 10 comes from 2 x 5
3. And we need to account for all the products of 5 and 2. likes 4×5 = 20 ...
4. So, if we take all the numbers with 5 as a factor, we'll have way more than enough even numbers to pair with them to get factors of 10

Example One

How many multiples of 5 are between 1 and 23? There is 5, 10, 15, and 20, for four multiples of 5. Paired with 2's from the even factors, this makes for four factors of 10, so:23! has 4 zeros.

Example Two

How many multiples of 5 are there in the numbers from 1 to 100?

because 100 ÷ 5 = 20, so, there are twenty multiples of 5 between 1 and 100.

but wait, actually 25 is 5×5, so each multiple of 25 has an extra factor of 5, e.g. 25 × 4 = 100，which introduces extra of zero.

So, we need know how many multiples of 25 are between 1 and 100? Since 100 ÷ 25 = 4, there are four multiples of 25 between 1 and 100.

Finally, we get 20 + 4 = 24 trailing zeroes in 100!

The above example tell us, we need care about 5, 5×5, 5×5×5, 5×5×5×5 ....

Example Three

By given number 4617.

5^1 : 4617 ÷ 5 = 923.4, so we get 923 factors of 5

5^2 : 4617 ÷ 25 = 184.68, so we get 184 additional factors of 5

5^3 : 4617 ÷ 125 = 36.936, so we get 36 additional factors of 5

5^4 : 4617 ÷ 625 = 7.3872, so we get 7 additional factors of 5

5^5 : 4617 ÷ 3125 = 1.47744, so we get 1 more factor of 5

5^6 : 4617 ÷ 15625 = 0.295488, which is less than 1, so stop here.

Then 4617! has 923 + 184 + 36 + 7 + 1 = 1151 trailing zeroes.

To avoid the integer overflow as @localvar mentioned below(in case of 'n >=1808548329' ), the expression " i <= INT_MAX/5" is not a good way to prevent overflow, because 5^13 is > INT_MAX/5 and it's valid.

So, if you want to use "multiply", consider define the 'i' as 'long long' type.

int trailingZeroes(int n) {
    int result = 0;
    for(long long i=5; n/i>0; i*=5){
        result += (n/i);
    }
    return result;
}