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 Harshad number or 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).

Title Harshad number HarshadNumber 2013-03-22 15:47:04 2013-03-22 15:47:04 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A63 Niven number Harshad number