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

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

Numbers of $5$ digits that have at least one of the digits repeated more than one time?

How can I count the numbers of $5$ digits such that at least one of the digits appears more than one time? My thoughts are: I count all the possible numbers of $5$ digits: $10^5 = 100000$. Then, I subtract the numbers that don’t have repeated digits, which I calculate this way: $10*9*8*7*6$ $= 30240 … 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