Articles of decimal expansion

Prove that it is impossible to have integers $b=5a$ under a digit re-ordering

Let $a$ be a positive integer and let $b$ be obtained from a by moving the initial digit of $a$ to the end. Prove that it is impossible to have $b=5a$.

What will be the units digit of $7777^{8888}$?

What will be the units digit of $7777$ raised to the power of $8888$ ? Can someone do the math with explaining the fact “units digit of $7777$ raised to the power of $8888$”?

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592…$, it would be the thousandth place, where the second $3$ is. To clarify, this notion of repetition means a pattern like abcdabcdefgh…

Representing negative numbers with an infinite number?

Motivation We all know that: $$ .\bar{9} =.999 \dots= 1$$ I was wondering if the following (obviously not rigorous) statement could be defined on the same footing? Question $$ x = \bar{9} $$ $$ \implies x/10 = \bar{9}.9 $$ Subtracting the above equations: $$ .9 x = -.9 $$ $$ \implies x = -1 $$ […]

Is there a number that is palindromal in both base 2 and base 3?

I manually checked the first 20 base 2 palindromes and I did not find any base 3 palindromes among them. Is there any definite way of determining this? What about other bases?

Last two digits of $17^{17^{17}}$

For a problem set, we had two find the final two digits of 17^17^17 So what I did was find the last digit of 17^17 and then take 17^of that last digit of 17 and then find the last two digits of that number. I got the last two digits as 17. Is my method […]

Doubling sequences of the cyclic decimal parts of the fraction numbers

Is there any theory, why and when doubling sequences of the decimal part of the fraction numbers occur? Take for example these small numbers: 1/7 = 0.[142857] 1/19 = 0.[052631578947368421] 1/49 = 0.[020408163265306122448979591836734693877551] where sequence inside [] is the repeating/cycling part. For 1/7 we can see that doubling is started with 14, following with 28, […]

Can permutating the digits of an irrational/transcendental number give any other such number?

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number’s sequence of decimal digits through a permutation of $x_n$?

How to know that irrational numbers never repeat?

How would you respond to a middle school student that says: “How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. And, how do they know that numbers like $\pi$ and $\sqrt2$ are irrational because they can’t check an infinite number digits […]

The number of ones in a binary representation of an integer

Is there any relation that tells whether the number of ones in a binary representation of an integer is an even or an odd number?