Problem:
What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
Solution:
The sum of nine consecutive integers is times the fifth number, the sum of ten consecutive integers is times the sum of the fifth and sixth numbers, and the sum of eleven consecutive integers is times the sixth number. Thus any positive integer that can be written as a sum of nine, ten, and eleven consecutive positive integers must be a multiple of and . The smallest such number is . It is readily verified that
The problems on this page are the property of the MAA's American Mathematics Competitions