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Non-negative Integers


Date: 11/15/2001 at 22:51:01
From: Jannet Stockton
Subject: Math

1) How many nonnegative integers consisting of one to three digits are 
   divisible by 5? Leading zeros are not allowed.

   I don't know that it includes 0 or not; if it includes 0, the 
   answer is 200.

2) How many nonnegative integers consisting of one to three different 
   digits are divisible by 5? Leading zeros are not allowed.

   The answer is 155, but how can I get it?


Date: 11/16/2001 at 16:30:16
From: Doctor Ian
Subject: Re: Math

Hi Jannet,

Zero is non-negative, so I would say it's included. 

For number 2, since you already know how many integers are divisible 
by 5, all you have to do is subtract the ones that have duplicate 
digits, e.g., 110, 225, 505, and so on. 

Apparently there are 45 such numbers, so you need to show that there 
is a systematic way to find them... i.e., one that is guaranteed to 
get all of them.

You could start by thinking about the patterns that the numbers would 
have to fall into to (1) be divisible by 5 and (2) have a duplicate 
digit.  Here are some:

  aa0     There would be 9 like this.
  aa5     And 9 more like this.
  5a5     And 9 more like this.
  a00     And 9 more like this.

That gives you 36. If you can find one more pattern, you'll have found 
them all. 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
High School Puzzles

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