hundrednumber

Hundrednumber is a term used in mathematics to denote a natural number that can be expressed as the sum of 100 consecutive integers. This class of numbers arises in problems about representing integers as sequences of consecutive terms.

If N is a hundrednumber, then there exists an integer a such that N = a + (a+1) + ... + (a+99). The sum of 100 consecutive integers equals 100a plus the sum of the first 99 nonnegative integers, which is 100a + 4950. Therefore N must satisfy N ≡ 50 (mod 100). Conversely, any integer N with N ≡ 50 (mod 100) can be written as a + (a+1) + ... + (a+99) with a = (N − 4950)/100. If we require all terms to be positive, then a ≥ 1, which yields N ≥ 5050; the positive hundrednumbers then form the increasing sequence 5050, 5150, 5250, 5350, and so on, each differing by 100.

This concept is a special case of the general question of representing numbers as sums of k consecutive integers. For a fixed even k, a natural number N can be expressed as a sum of k consecutive integers if and only if N ≡ k/2 (mod k). In the case k = 100, this condition becomes N ≡ 50 (mod 100). The hundrednumber framework highlights the modular constraint that governs such representations.

The term hundrednumber is not a standard mathematical term, but it provides a concise label in problem statements and expository discussions about sums of consecutive integers.