## Is the last digit of this number :44n+1 {{4^4}^n}+1 always 77 for n>1n>1 and could this be prime?

Some computations in wolfram alpha for n=2,3,4,5,6 showed that the last digit of this number 44n+1 for n>1 always 7 . My question here :How do I know if it’s last digit always is 7 ? Note: My Goal is to know for which values of n: 44n+1 could be prime ? Thank you for … Read more

## Find $abcd$ x 4 = $dcba$

Find $abcd$ x 4 = $dcba$ And in general, find: $abcd$ x $e$ = $dcba$ This was a problem intended for 3rd graders I believe. I’m having some trouble breaking down an intuitive method that a young student can understand. Answer First, answer for the specific problem. $d\geq 4$ because $dcba=4\times abcd\geq 4\times 1000=4000$. Moreover, … Read more

## Analysis of a proof that the decimal expression for any rational is periodic.

Martin Liebeck in his book “A Concise Introduction to Pure Mathematics” (Third edition) writes the following proposition and proof (Chapter 3, pg 24) PROPOSITION 3.4 The decimal expression for any rational number is periodic PROOF Consider the rational $m\over{n}$ (Where m, n, $\in$ $\mathbb{Z}$). To express this as a decimal, we perform long division of … Read more

## Does 1/3 have a unique decimal representation?

I think it does, but I’m not sure. And also there are rationals which have unique decimal representation besides irrational numbers. Am i right? Answer Yes, you are right. The only real numbers with more than one decimal representations are those that can be written as $\frac k{10^n}$, with $k\in\mathbb Z\setminus\{0\}$ and $n\in\{0,1,2,\ldots\}$. Those that … Read more

## How do I properly say .999…?

I am making a video for something and I was wondering what the most optimal way to say .999…? I don’t want to keep on saying .999 repeated, so what should I be saying? Edit: I can’t call it 1 as that is what the video is about(proving .999…=1)! Answer Zero point nine recurring. Although … Read more

## A 1717-digit number and the number formed by reversing its digits are added together. Show that the sum has a even digit.

A 17 digit number is chosen, and it’s digits are reversed, forming a new number. These two numbers are added together. Show that there sum has at least one even digit. The solution given in the book is as follows: Suppose there were a 17-digit integer whose reversed sum contained no even digit. For convenience, … Read more

## Show that the decimal number obtained by concatenating the digits of n! successively represents an irrational number.

Show that the decimal number 0.12624120720… obtained by concatenating the digits of n! successively with n=1,2,3,… represents an irrational number. A rational number either has a terminating decimal expansion or an eventually repeating decimal expansion. 0.12624120720… is clearly not a terminating sequence. n! is unique for each n∈N But how do we prove that uniqueness … Read more