3
votes
5answers
578 views

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 ...
-1
votes
3answers
111 views

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 $m$.
1
vote
2answers
30 views

$3^a\mid s(n) \Rightarrow 3^a\mid n$

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 ...
1
vote
1answer
26 views

Proof that there are at the most two numbers of exactly six digits that squared end with the same six digits

Written in a more formal way, proof that there are at the most $2$ numbers $n$ of six digits, that $$n^2 \equiv n \mod 10^6$$ Research effort: if $n^2 \equiv n \mod 10^6$ this means $10^6\mid ...
0
votes
0answers
61 views

Natural numbers, a proof for the divisibility of any 3 given numbers?

I'm following EdX "Effective Thinking Through Mathematics" and they posed the following question: "If $x, y, z$ are natural numbers other than 1, and you multiply them together and add 1, ($x ...
1
vote
1answer
80 views

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 ...
1
vote
1answer
47 views

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 ...
3
votes
3answers
485 views

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 ...
1
vote
4answers
383 views

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, ...
4
votes
5answers
442 views

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 ...
2
votes
1answer
163 views

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 $1978^N - 1$. ...