Three Number Theory QuestionsDate: 10/25/1999 at 09:09:46 From: Christopher Subject: Number Theory FIRST QUESTION: 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. SECOND QUESTION: How many times does the digit 1 occur from 1 up to 10,000,000,000? Example: from 1 to 11: 4 times THIRD QUESTION: 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 FIRST QUESTION: 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?) SECOND QUESTION: 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. THIRD QUESTION: 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 http://mathforum.org/dr.math/ |
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