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

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…

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: $$ .9 x = -.9 $$ $$ \implies x = -1 $$ […]

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?

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 […]

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, […]

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 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 […]

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?

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