It has just occurred to me that there is a very simple test to check if an integer is divisible by 4: take twice its tens place and add it to its ones place. If that number is divisible by 4, so is ...
Does this have a name: If an odd prime $p$ does not divide $a$, then $p$ divides $a^n + 1$ or $a^n - 1$
After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in ...
For a prime $p$ and positive integer $n$, let $E(n,p)$ be the greatest $k$ such that $p^k \mid n$, and $E(n,p) = 0$ if $p \nmid n$. Let $E(n) = E(n, 2)$. A number of years back, I proved the ...
I'm searching for a good book or web page that has a good amount of problems and their solutions, at undergraduate level, of divisibility, inequalities, induction, etc. Thanks in advance