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Lucky Number Sequences

Date: 05/11/98 at 05:54:10
From: Matt Dellit
Subject: Lucky Number

A lucky number is one for which the sum of its digits is divisible by 
7. For example, 7, 25 and 849 are lucky numbers. The smallest pair of 
lucky numbers is 7 and 16; they differ by 9.

The problems:

a) Find eight consecutive numbers two of which are lucky.

b) Find 12 consecutive numbers none of which is lucky.

c) Show that 13 consecutive numbers always contain at least one 
   lucky number.

d) "Hey," says Chris, "I've found the largest pair of lucky numbers 
   that differ by 9!" Paris replies, "Chris, you're wrong!" Explain 
   how Paris knows Chris is wrong.

Thanks for this.

Matt Dellit

Date: 05/12/98 at 13:36:56
From: Doctor Dennis
Subject: Re: Lucky Number

For problem a, a little experimentation leads to the answer 160-167 
(160 and 167 are lucky). Basically, my idea was to just try bigger 
numbers until finding an answer. Also, realizing that numbers with 
digit sum 7 must be at least 9 apart.

For problem b, I found 994-1005, using basically more guesswork. If 
you play with the numbers for a while, it becomes clear that most of 
the time there is a lucky number every 9 numbers, so to get every 12 
we need to find a "strange" area in terms of the sum of digits. One 
such strange area is the 999-1000 line, where the digit sum changes 
drastically. Probably there are many other similar areas, but I 
couldn't think of any with 2 or fewer digits.

In problem c, each time we change the ones digit only, we increase the 
sum, and hence the remainder of division by 7, by 1. The remainder of 
division by 7 can only be 1 through 6 (0 means we have a lucky number,
which we don't want). Also, within 13 consecutive numbers we can only 
change digits other than the ones digit once (meaning we carried to 
the next digit), but when we change other digits we can reset the 
remainder of division by 7. For example, using the numbers from 
problem b:

   number    remainder when we divide by 7
    994                  1
    995                  2
    996                  3
    997                  4
    998                  5
    999                  6
   1000                  1
   1001                  2
   1002                  3
   1003                  4
   1004                  5
   1005                  6

And we find 12 numbers in a row that are not lucky numbers. Thinking 
about all this, can you see why 13 numbers must contain one lucky 
number? The basic idea here is to try to construct as long a sequence 
as possible and show that it can only have 12 numbers without a 
lucky number. 

For problem d, consider the pair 151 and 160. These differ by 9. So do 
1051 and 1060. As do 10051 and 10060. And 10000000000000051 and 
10000000000000060. This should set off some ideas about why Chris 
is wrong.

Let me know if you have any more questions. 

-Doctor Dennis,  The Math Forum
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Associated Topics:
High School Number Theory

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