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Three Number Theory Questions

Date: 10/25/1999 at 09:09:46
From: Christopher 
Subject: Number Theory


Given: 4444^4444

Question: The summation of the digits of the value of the given number 
is represented as A. The summation of the digits of A is represented 
as B. Find the summation of the digits of B.


How many times does the digit 1 occur from 1 up to 10,000,000,000?
Example: from 1 to 11: 4 times


Given: (5^1985)-1

Question: Find 3 integers greater than 5^100 that are factors of the 
given number.

Date: 10/25/1999 at 17:52:41
From: Doctor Schwa
Subject: Re: Number Theory


Cool question! Two ideas to use here:

1) When you sum the digits, the remainder mod 9 doesn't change, so 
   find the remainder of 4444^4444 / 9.

2) How big can it be? (How many digits does the original number have? 
   So what's the biggest A can be? ...B?  ...the final answer?)


This seems more like a counting question than a number theory 
question. I would approach it as

     (1 * number of numbers that have exactly one 1) +
     (2 * number of numbers that have exactly two 1s) +
     (3 * number of numbers that have exactly three 1s) +
     and so on.


Remember that x^n - 1 is always divisible by x - 1. So for example, 
3^20 - 1 is divisible by 3 - 1. But besides that, you can rewrite it:

     3^20 - 1 = (3^4)^5 - 1

and now you can think of the 3^4 as the x, and so it is divisible by 
3^4 - 1.

So now you have a small (in the case of problem 2), medium (in the 
case of problem 1), or large (in the case of problem 3) hint. See how 
far you can get with those and please write back if you'd like a 
slightly larger hint!

- Doctor Schwa, The Math Forum   
Associated Topics:
High School Discrete Mathematics
High School Number Theory

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