# Tagged Questions

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### Determine the divisibility of a given number without performing full division

My question is slightly more complicated than what's implied on the title, so I will start with an example. Given any number $N$ on base $10$, we can easily determine whether or not $N$ is divisible ...
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### The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each ...
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### Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
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### $3^a\mid s(n) \Rightarrow 3^a\mid n$

This is not a homework question, neither a championship problem (as far as I've searched in the net), and it came up noticing a singular pattern, involving the powers of $3$: "Prove or disprove that ...
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### Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since ...
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### Pigeonhole question - divisibility by chosen number

This question should be solved with pigeonhole principle. Let $a,n \in \mathbb N$ such that $a$ is a number whose digits are only $3$'s and $0$'s, and $n$ is an unspecified natural number. Show that ...
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### How to divide by 12 quickly?

Let $n\in\mathbb N$ be divisible by 12 and $n/12<100$. Is there a way of computing $n/12$ rather quickly using mental arithmetic (e.g. for 972/12, 1044/12, etc.)? For example, the number 11 seems ...
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### Does such a natural number exist, that it would be divisible by every other natural number

I've got to prove (or disprove) the following statement: $\exists x \in \mathbb{N} \; \forall y \in \mathbb{N}: y \mid x$, which translates into "It exists such $x$ from the set of natural numbers, ...
### Prove 24 divides $u^3-u$ for all odd natural numbers $u$
At our college, a professor told us to prove by a semi-formal demonstration (without complete induction): For every odd natural: $24\mid(u^3-u)$ He said that that example was taken from a high ...
### multiple approaches/ways to prove that $1000^N - 1$ cannot be a divisor of $1978^N - 1$
Am interested in learning to do multiple proofs for the same problem, and hence I chose this problem: Prove that for any natural number $N$, $1000^N - 1$ cannot be a divisor of $1978^N - 1$. ...