SEARCH HOME
Math CentralQuandaries & Queries

search

Question from Ben, a teacher:

Question: Using mathematical induction, prove that if the sum of the digits of a number is divisible by three, then the number itself is also divisible by 3.

Ben,

You need to think about what a number like 9826 means.

9823 = 9(10^3) + 8(10^2) + 2(10) + 6 = 9(999+1) + 8(99+1) + 2(9+1) + 6 = 9(999) + 8(99) + 2(9) + 9 + 8 + 2 + 6,

now clearly all the terms with all the '9's' in them are a multiple of 3 so the original number is a multiple of 3 if and only if 9 + 8 + 2 + 6 is a multiple of 3.

To use induction you would induct on the number of digits - at the induction step you would pull off the 1st digit and it accompanying power of 10 and apply the induction hypothesis to the rest.

Penny

 

About Math Central
 

 


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS