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 = $dcba$ 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
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
In the text TAOCP A real number is a quantity x that has a decimal expansion x=n+0.d1d2d3… where n is an integer, each di is a digit between 0 and 9, and the sequence of digits doesn’t end with infinitely many 9s Why is such a case ? Isn’t all the numbers between 0 and … Read more
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
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 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
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
What is this question asking? Do you just simply count to number of 1’s and 0’s and then choose that integer, 6? y=10k+10k+1+10k+2 In the equation above, k is an integer. For what value of k can y be expressed as an integer whose digits consist of twice as many 0‘s as 1‘s? (original photo … Read more
I know this would be kind of silly, but then this has been troubling me for the past few days. All of us know $2^{\frac{1}{2}}$ is irrational. Let us try to mark this on the number line “exactly”, as in trying to take the first $100$ decimal places, and try to mark it through successive … Read more
