trailingzero

Trailingzero is a term used in mathematics, particularly in number theory and combinatorics, to refer to the number of zero digits at the end of a number expressed in a given base. Most commonly, it refers to the number of trailing zeros in the decimal representation (base 10) of an integer.

The presence of trailing zeros in a base-10 number indicates divisibility by powers of 10. Since 10 is the product of 2 and 5, trailing zeros are directly related to the factors of 2 and 5 in the prime factorization of the number. Specifically, each trailing zero corresponds to one factor of 10, which is equivalent to one pair of 2 and 5.

A common problem involving trailing zeros is determining the number of trailing zeros in the factorial of a non-negative integer, denoted as n!. For example, 5! = 120, which has one trailing zero. 10! = 3,628,800, which has two trailing zeros.

To find the number of trailing zeros in n!, one needs to count the number of times 10 is a factor in its prime factorization. This is equivalent to counting the number of pairs of 2 and 5. Since the factor of 2 appears more frequently than the factor of 5 in the prime factorization of n!, the number of trailing zeros is determined by the count of the factor of 5. This count can be calculated by summing the quotients of n divided by successive powers of 5: floor(n/5) + floor(n/25) + floor(n/125) + ... until the power of 5 exceeds n. This formula is often attributed to Legendre.

The concept of trailing zeros can be extended to other bases. For instance, in base 2 (binary), trailing zeros indicate divisibility by powers of 2. The number of trailing zeros in n! in base b is determined by the number of times the prime factors of b appear in the prime factorization of n!, prioritizing the least frequent prime factor of b.