trailingzeros

Trailing zeros are the zeros at the end of the decimal representation of a number. They occur when the number is divisible by powers of the base; in decimal, they correspond to factors of 10 in the number. The number of trailing zeros of an integer n in base 10 equals the largest k such that 10^k divides n, which is equivalently the minimum of the exponents of 2 and 5 in the prime factorization of n (v2(n) and v5(n)). For example, 2500 ends with two zeros because 2500 = 2^2 · 5^4, so min(2,4) = 2.

Trailing zeros are particularly studied for factorials. The number of trailing zeros in n! is the sum floor(n/5) + floor(n/25) + floor(n/125) + ... until the powers of 5 exceed n. This accounts for multiples of 5, 25, 125, etc., which contribute additional zeros. For instance, 10! = 3628800 has two trailing zeros, since floor(10/5) = 2 and floor(10/25) = 0.

More generally, in any base b with prime factorization b = p1^a1 p2^a2 ..., a number n has k trailing zeros in base b if and only if n is divisible by b^k, i.e., k = min_i floor(v_{pi}(n) / ai). In decimal, b = 10 = 2 · 5, so k = min(v2(n), v5(n)).

Trailing zeros have applications in number theory, combinatorics, and programming, where they indicate divisibility properties and are used in counting problems and algorithmic tasks.