HarshadNumber 2006-03-18 16:38:21 2006-06-13 11:24:47 Definition Harshad number % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Consider the integer 1729. Adding up its digits, $$1 + 7 + 2 + 9 = 19$$ and $${{1729} \over {19}} = 91.$$ When an integer is divisible by the sum of its digits, it's called a {\em Harshad number} or {\em Niven number}. That is, given m is the number of digits of n and d is an integer of n, $${\sum_{i = 1}^m d_i}|n$$ All 1-digit numbers and the base number itself are Harshad numbers. 1, 2, 4 and 6 are always Harshad numbers regardless of the base. It is possible for an integer to be divisible by its digital root and yet not be a Harshad number because it doesn't divide its first digit sum evenly (for example, 38 in base 10 has digital root 2 but is not divisible by 3 + 8 = 11). The reverse is also possible (for example, 195 is divisible by 1 + 9 + 5 = 15, but not by its digital root 4).